Glossaries of mathematicsFrom Wikipedia, the free encyclopedia

Contents

1 Glossary of areas of mathematics 11.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.10 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.11 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.12 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.13 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.14 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.15 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.16 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.17 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.18 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.19 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.20 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.21 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.22 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.23 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.24 X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.25 Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.26 Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.27 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Glossary of arithmetic and Diophantine geometry 192.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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2.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.10 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.11 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.12 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.13 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.14 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.15 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.16 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.17 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.18 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.19 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.20 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.21 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.22 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.24 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Glossary of classical algebraic geometry 303.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.7 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.8 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.9 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.10 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.11 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.12 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.13 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.14 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.15 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.16 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.17 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.18 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.19 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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3.20 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.21 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.22 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.23 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.24 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.25 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.26 XYZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.27 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Glossary of cryptographic keys 544.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Glossary of differential geometry and topology 575.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.10 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.11 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.12 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.13 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.14 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.15 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.16 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.17 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 Glossary of experimental design 616.1 Concerned fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7 Glossary of field theory 657.1 Definition of a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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7.4 Types of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.5 Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.6 Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.7 Extensions of Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8 Glossary of game theory 718.1 Definitions of a game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8.1.1 Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.1.2 Normal form game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.1.3 Extensive form game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728.1.4 Cooperative game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728.1.5 Simple game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.2 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

9 Glossary of graph theory 759.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

9.1.1 Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779.1.2 Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789.1.3 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799.1.4 Cliques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799.1.5 Strongly connected component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809.1.6 Hypercubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819.1.7 Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819.1.8 Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819.1.9 Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

9.2 Adjacency and degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829.2.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

9.3 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839.4 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839.5 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849.6 Genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849.7 Weighted graphs and networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859.8 Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

9.8.1 Directed acyclic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869.9 Colouring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869.10 Various . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

10 Glossary of invariant theory 8910.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

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10.2 !$@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9010.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9010.4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110.5 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110.6 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9310.7 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9310.8 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9410.9 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9410.10H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9410.11I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9510.12J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9510.13K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9510.14L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9610.15M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9610.16N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9610.17O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9610.18P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9710.19Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9710.20R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9810.21S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9910.22T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10010.23U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10010.24V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10010.25W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10010.26XYZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.27See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.29External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

11 Glossary of Lie algebras 10311.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

11.1.1 subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10411.2 Solvability, nilpotency, Jordan decomposition, semisimplicity . . . . . . . . . . . . . . . . . . . . 10411.3 Semisimple Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10511.4 Root System (for classification of semisimple Lie algebra) . . . . . . . . . . . . . . . . . . . . . . 105

11.4.1 theory of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10711.5 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

11.5.1 Universal enveloping algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10811.5.2 Verma modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

11.6 cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10811.7 Chevalley basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10811.8 Examples of Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

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11.8.1 complex Lie algebras of 1D, 2D, 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10911.8.2 Simple Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10911.8.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

11.9 Other discipline related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10911.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

12 Glossary of module theory 11012.1 Basic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11012.2 Types of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11112.3 Operations on modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

12.3.1 Changing scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11412.4 Homological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11412.5 Modules over special rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11412.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11412.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11412.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

13 Glossary of order theory 11613.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11713.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11713.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11813.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11813.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11913.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11913.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11913.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11913.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12013.10J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12013.11L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12013.12M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12113.13O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12113.14P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12113.15Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12213.16R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12213.17S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12213.18T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12313.19U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12313.20V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12313.21W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12313.22Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12413.23Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12413.24References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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14 Glossary of Riemannian and metric geometry 12514.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12614.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12614.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12714.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12714.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12714.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12714.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12814.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12814.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12814.10J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12814.11K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12814.12L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12914.13M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12914.14N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12914.15P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12914.16Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12914.17R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13014.18S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13014.19T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13114.20U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13114.21W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

15 Glossary of ring theory 13215.1 Definition of a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13215.2 Types of elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13215.3 Homomorphisms and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13315.4 Types of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13415.5 Ring constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

15.5.1 Polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13615.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13715.7 Ringlike structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13715.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13715.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13715.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

16 Glossary of semisimple groups 13816.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13916.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13916.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13916.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14016.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

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16.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14016.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14016.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14116.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14116.10J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14116.11K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14116.12L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14116.13M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14116.14N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14116.15O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14216.16P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14216.17Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14216.18R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14216.19S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14216.20T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14216.21U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14216.22V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14316.23W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

17 Glossary of shapes with metaphorical names 14417.1 Numbers and letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14517.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14717.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

18 Glossary of string theory 14818.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14918.2 αβγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14918.3 !$@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15118.4 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15118.5 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15218.6 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15218.7 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15418.8 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15518.9 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15618.10G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15618.11H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15718.12I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15818.13J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15818.14K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15818.15L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15918.16M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16018.17N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

CONTENTS ix

18.18O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16218.19P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16218.20Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16318.21R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16318.22S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16418.23T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16518.24U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16618.25V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16618.26W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16718.27XYZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16718.28See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16718.29References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16818.30External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

19 Glossary of systems theory 16919.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17019.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17019.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17019.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17019.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17119.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17119.7 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17119.8 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17219.9 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17219.10M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17219.11O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17219.12P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17219.13R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17219.14S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17319.15W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17419.16See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17519.17References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17519.18External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

20 Glossary of tensor theory 17620.1 Classical notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17620.2 Algebraic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

20.2.1 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17720.2.2 Pure tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17720.2.3 Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17820.2.4 Hodge star operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17820.2.5 Exterior power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

x CONTENTS

20.2.6 Symmetric power, symmetric algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17820.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17820.4 Tensor field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17820.5 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17820.6 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17920.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17920.8 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

21 Glossary of topology 18121.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18221.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18321.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18321.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18521.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18521.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18521.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18621.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18621.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18721.10K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18721.11L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18821.12M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18821.13N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18921.14O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19021.15P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19021.16Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19121.17R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19221.18S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19221.19T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19321.20U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19421.21W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19521.22Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19521.23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19521.24External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

22 List of mathematical jargon 19722.1 Philosophy of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19722.2 Descriptive informalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19922.3 Proof terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20122.4 Proof techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20222.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20322.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20322.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 204

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22.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20422.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20522.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Chapter 1

Glossary of areas of mathematics

This is a glossary of terms that are or have been considered areas of study in mathematics.

1.1 A

• Absolute differential calculus— the original name for tensor calculus developed around 1890.

• Absolute geometry — an extension of ordered geometry that is sometimes referred to as neutral geometrybecause its axiom system is neutral to the parallel postulate.

• Abstract algebra— the study of algebraic structures and their properties. Originally it was known as modernalgebra.

• Abstract analytic number theory — a branch mathematics that take ideas from classical analytic numbertheory and applies them to various other areas of mathematics.

• Abstract differential geometry — a form of differential geometry without the notion of smoothness fromcalculus. Instead it is built using sheaf theory and sheaf cohomology.

• Abstract harmonic analysis — a modern branch of harmonic analysis that extends upon the generalizedFourier transforms that can be defined on locally compact groups.

• Abstract homotopy theory

• Additive combinatorics — the part of arithmetic combinatorics devoted to the operations of addition andsubtraction.

• Additive number theory—a part of number theory that studies subsets of integers and their behaviour underaddition.

• Affine geometry— a branch of geometry that is centered on the study of geometric properties that remainunchanged by affine transformations. It can be described as a generalization of Euclidean geometry.

• Affine geometry of curves— the study of curves in affine space.

• Affine differential geometry — a type of differential geometry dedicated to differential invariants undervolume-preserving affine transformations.

• Ahlfors theory— a part of complex analysis being the geometric counterpart of Nevanlinna theory. It wasinvented by Lars Ahlfors

• Algebra—amajor part of pure mathematics centered on operations and relations. Beginning with elementaryalgebra, it introduces the concept of variables and how these can be manipulated towards problem solving;known as equation solving. Generalizations of operations and relations defined on sets have led to the ideaof an algebraic structure which are studied in abstract algebra. Other branches of algebra include universalalgebra, linear algebra and multilinear algebra.

1

2 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS

• Algebraic analysis—motivated by systems of linear partial differential equations, it is a branch of algebraicgeometry and algebraic topology that uses methods from sheaf theory and complex analysis, to study the prop-erties and generalizations of functions. It was started by Mikio Sato.

• Algebraic combinatorics— an area that employs methods of abstract algebra to problems of combinatorics.It also refers to the application of methods from combinatorics to problems in abstract algebra.

• Algebraic computation— see symbolic computation.

• Algebraic geometry— a branch that combines techniques from abstract algebra with the language and prob-lems of geometry. Fundamentally, it studies algebraic varieties.

• Algebraic graph theory— a branch of graph theory in which methods are taken from algebra and employedto problems about graphs. The methods are commonly taken from group theory and linear algebra.

• Algebraic K-theory — an important part of homological algebra concerned with defining and applying acertain sequence of functors from rings to abelian groups.

• Algebraic number theory— a part of algebraic geometry devoted to the study of the points of the algebraicvarieties whose coordinates belong to an algebraic number field. It is a major branch of number theory and isalso said to study algebraic structures related to algebraic integers.

• Algebraic statistics— the use of algebra to advance statistics, although the term is sometimes restricted tolabel the use of algebraic geometry and commutative algebra in statistics.

• Algebraic topology— a branch that uses tools from abstract algebra for topology to study topological spaces.

• Algorithmic number theory— also known as computational number theory, it is the study of algorithms forperforming number theoretic computations.

• Anabelian geometry — an area of study based on the theory proposed by Alexander Grothendieck in the1980s that describes the way a geometric object of an algebraic variety (such as an algebraic fundamentalgroup) can be mapped into another object, without it being an abelian group.

• Analysis— a rigorous branch of pure mathematics that had its beginnings in the formulation of infinitesimalcalculus. (Then it was known as infinitesimal analysis.) The classical forms of analysis are real analysis and itsextension complex analysis, whilst more modern forms are those such as functional analysis.

• Analytic combinatorics—part of enumerative combinatorics where methods of complex analysis are appliedto generating functions.

• Analytic geometry — usually this refer to the study of geometry using a coordinate system (also knownas Cartesian geometry). Alternatively it can refer to the geometry of analytic varieties. In this respect it isessentially equivalent to real and complex algebraic geometry.

• Analytic number theory—part of number theory using methods of analysis (as opposed to algebraic numbertheory)

• Appliedmathematics—acombination of various parts ofmathematics that concern a variety ofmathematicalmethods that can be applied to practical and theoretical problems. Typically the methods used are for science,engineering, finance, economics and logistics.

• Approximation theory — part of analysis that studies how well functions can be approximated by simplerones (such as polynomials or trigonometric polynomials)

• Arakelov geometry— also known as Arakelov theory

• Arakelov theory — an approach to Diophantine geometry used to study Diophantine equations in higherdimensions (using techniques from algebraic geometry). It is named after Suren Arakelov.

• Arithmetic—tomost people this refers to the branch known as elementary arithmetic dedicated to the usage ofaddition, subtraction, multiplication and division. However arithmetic also includes higher arithmetic referringto advanced results from number theory.

• Arithmetic algebraic geometry— see arithmetic geometry

1.2. B 3

• Arithmetic combinatorics—the study of the estimates from combinatorics that are associated with arithmeticoperations such as addition, subtraction, multiplication and division.

• Arithmetic dynamics

• Arithmetic geometry— the study of schemes of finite type over the spectrum of the ring of integers

• Arithmetic topology— a combination of algebraic number theory and topology studying analogies betweenprime ideals and knots

• Arithmetical algebraic geometry— an alternative name for arithmetic algebraic geometry

• Asymptotic combinatorics

• Asymptotic geometric analysis

• Asymptotic theory— the study of asymptotic expansions

• Auslander–Reiten theory— the study of the representation theory of Artinian rings

• Axiomatic geometry— see synthetic geometry.

• Axiomatic homology theory

• Axiomatic set theory— the study of systems of axioms in a context relevant to set theory and mathematicallogic.

1.2 B• Bifurcation theory— the study of changes in the qualitative or topological structure of a given family. It is apart of dynamical systems theory

• Birational geometry— a part of algebraic geometry that deals with the geometry (of an algebraic variety)that is dependent only on its function field.

• Bolyai-Lobachevskian geometry— see hyperbolic geometry.

1.3 C• C*-algebra theory

• Cartesian geometry— see analytic geometry

• Calculus—a branch usually associated with limits, functions, derivatives, integrals and infinite series. It formsthe basis of classical analysis, and historically was called the calculus of infinitesimals or infinitesimal calculus.Now it can refer to a system of calculation guided by symbolic manipulation.

• Calculus of infinitesimals— also known as infinitesimal calculus. It is a branch of calculus built upon theconcepts of infinitesimals.

• Calculus of moving surfaces—an extension of the theory of tensor calculus to include deforming manifolds.

• Calculus of variations— the field dedicated to maximizing or minimizing functionals. It used to be calledfunctional calculus.

• Catastrophe theory— a branch of bifurcation theory from dynamical systems theory, and also a special caseof the more general singularity theory from geometry. It analyses the germs of the catastrophe geometries.

• Categorical logic—a branch of category theory adjacent to the mathematical logic. It is based on type theoryfor intuitionistic logics.

• Category theory— the study of the properties of particular mathematical concepts by formalising them ascollections of objects and arrows.

4 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS

• Chaos theory — the study of the behaviour of dynamical systems that are highly sensitive to their initialconditions.

• Character theory— a branch of group theory that studies the characters of group representations or modularrepresentations.

• Class field theory— a branch of algebraic number theory that studies abelian extensions of number fields.

• Classical differential geometry— also known as Euclidean differential geometry. see Euclidean differentialgeometry.

• Classical algebraic topology

• Classical analysis—usually refers to the more traditional topics of analysis such as real analysis and complexanalysis. It includes any work that does not use techniques from functional analysis and is sometimes calledhard analysis. However it may also refer to mathematical analysis done according to the principles of classicalmathematics.

• Classical analytic number theory

• Classical differential calculus

• Classical Diophantine geometry

• Classical Euclidean geometry— see Euclidean geometry

• Classical geometry—may refer to solid geometry or classical Euclidean geometry. See geometry

• Classical invariant theory — the form of invariant theory that deals with describing polynomial functionsthat are invariant under transformations from a given linear group.

• Classical mathematics— the standard approach to mathematics based on classical logic and ZFC set theory.

• Classical projective geometry

• Classical tensor calculus

• Clifford analysis— the study of Dirac operators and Dirac type operators from geometry and analysis usingclifford algebras.

• Clifford theory is a branch of representation theory spawned from Cliffords theorem.

• Cobordism theory

• Cohomology theory

• Combinatorial analysis

• Combinatorial commutative algebra—a discipline viewed as the intersection between commutative algebraand combinatorics. It frequently employsmethods from one to address problems arising in the other. Polyhedralgeometry also plays a significant role.

• Combinatorial design theory— a part of combinatorial mathematics that deals with the existence and con-struction of systems of fintie sets whose intersections have certain properties.

• Combinatorial game theory

• Combinatorial geometry— see discrete geometry

• Combinatorial group theory— the theory of free groups and the presentation of a group. It is closely relatedto geometric group theory and is applied in geometric topology.

• Combinatorial mathematics

• Combinatorial number theory

• Combinatorial set theory— also known as Infinitary combinatorics. see infinitary combinatorics

1.3. C 5

• Combinatorial theory

• Combinatorial topology— an old name for algebraic topology, when topological invariants of spaces wereregarded as derived from combinatorial decompositions.

• Combinatorics — a branch of discrete mathematics concerned with countable structures. Branches of itinclude enumerative combinatorics, combinatorial design theory, matroid theory, extremal combinatorics andalgebraic combinatorics, as well as many more.

• Commutative algebra— a branch of abstract algebra studying commutative rings.

• Complex algebra

• Complex algebraic geometry— the main stream of algebraic geometry devoted to the study of the complexpoints of algebraic varieties.

• Complex analysis— a part of analysis that deals with functions of a complex variable.

• Complex analytic dynamics— a subdivision of complex dynamics being the study of the dynamic systemsdefined by analytic functions.

• Complex analytic geometry— the application of complex numbers to plane geometry.

• Complex differential geometry— a branch of differential geometry that studies complex manifolds.

• Complex dynamics — the study of dynamical systems defined by iterated functions on complex numberspaces.

• Complex geometry—the study of complexmanifolds and functions of complex variables. It includes complexalgebraic geometry and complex analytic geometry.

• Complexity theory— the study of complex systems with the inclusion of the theory of complex systems.

• Computable analysis— the study of which parts of real analysis and functional analysis can be carried outin a computable manner. It is closely related to constructive analysis.

• Computable model theory— a branch of model theory dealing with the relevant questions computability.

• Computability theory—abranch ofmathematical logic originating in the 1930s with the study of computablefunctions and Turing degrees, but now includes the study of generalized computability and definability. Itoverlaps with proof theory and effective descriptive set theory.

• Computational algebraic geometry

• Computational complexity theory—a branch of mathematics and theoretical computer science that focuseson classifying computational problems according to their inherent difficulty, and relating those classes to eachother.

• Computational geometry

• Computational group theory— the study of groups by means of computers.

• Computational mathematics — the mathematical research in areas of science where computing plays anessential role.

• Computational number theory— also known as algorithmic number theory, it is the study of algorithms forperforming number theoretic computations.

• Computational real algebraic geometry

• Computational synthetic geometry

• Computational topology

• Computer algebra— see symbolic computation

• Conformal geometry— the study of conformal transformations on a space.

6 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS

• Constructive analysis—mathematical analysis done according to the principles of constructive mathematics.This differs from classical analysis.

• Constructive function theory—a branch of analysis that is closely related to approximation theory, studyingthe connection between the smoothness of a function and its degree of approximation

• Constructive mathematics—mathematics which tends to use intuitionistic logic. Essentially that is classicallogic but without the assumption that the law of the excluded middle is an axiom.

• Constructive quantum field theory — a branch of mathematical physics that is devoted to showing thatquantum theory is mathematically compatible with special relativity.

• Constructive set theory

• Contact geometry — a branch of differential geometry and topology, closely related to and considered theodd-dimensional counterpart of symplectic geometry. It is the study of a geometric structure called a contactstructure on a differentiable manifold.

• Convex analysis— the study of properties of convex functions and convex sets.

• Convex geometry— part of geometry devoted to the study of convex sets.

• Coordinate geometry— see analytic geometry

• CR geometry— a branch of differential geometry, being the study of CR manifolds.

1.4 D• Derived noncommutative algebraic geometry

• Descriptive set theory— a part of mathematical logic, more specifically a part of set theory dedicated to thestudy of Polish spaces.

• Differential algebraic geometry—the adaption of methods and concepts from algebraic geometry to systemsof algebraic differential equations.

• Differential calculus— a subfield of calculus concerned with derivatives or the rates that quantities change.It is one of two traditional divisions of calculus, the other being integral calculus.

• Differential Galois theory— the study of the Galois groups of differential fields.

• Differential geometry— a form of geometry that uses techniques from integral and differential calculus aswell as linear and multilinear algebra to study problems in geometry. Classically, these were problems ofEuclidean geometry, although now it has been expanded. It is generally concerned with geometric structureson differentiable manifolds. It is closely related to differential topology.

• Differential geometry of curves— the study of smooth curves in Euclidean space by using techniques fromdifferential geometry

• Differential geometry of surfaces — the study of smooth surfaces with various additional structures usingthe techniques of differential geometry.

• Differential topology— a branch of topology that deals with differentiable functions on differentiable mani-folds.

• Diffiety theory

• Diophantine geometry— in general the study of algebraic varieties over fields that are finitely generated overtheir prime fields.

• Discrepancy theory

• Discrete computational geometry

• Discrete differential geometry

1.5. E 7

• Discrete dynamics

• Discrete exterior calculus

• Discrete geometry

• Discrete mathematics

• Discrete Morse theory— a combinatorial adaption of Morse theory.

• Distance geometry

• Domain theory

• Donaldson theory— the study of smooth 4-manifolds using gauge theory.

• Dynamical systems theory

1.5 E• Econometrics— the application of mathematical and statistical methods to economic data.

• Effective descriptive set theory— a branch of descriptive set theory dealing with set of real numbers thathave lightface definitions. It uses aspects of computability theory.

• Elementary algebra — a fundamental form of algebra extending on elementary arithmetic to include theconcept of variables.

• Elementary arithmetic— the simplified portion of arithmetic considered necessary for primary education.It includes the usage addition, subtraction, multiplication and division of the natural numbers. It also includesthe concept of fractions and negative numbers.

• Elementary mathematics—parts of mathematics frequently taught at the primary and secondary school lev-els. This includes elementary arithmetic, geometry, probability and statistics, elementary algebra and trigonometry.(calculus is not usually considered a part)

• Elementary group theory— the study of the basics of group theory

• Elimination theory— the classical name for algorithmic approaches to eliminating between polynomials ofseveral variables. It is a part of commutative algebra and algebraic geometry.

• Elliptic geometry— a type of non-Euclidean geometry (it violates Euclid's parallel postulate) and is basedon spherical geometry. It is constructed in elliptic space.

• Enumerative combinatorics — an area of combinatorics that deals with the number of ways that certainpatterns can be formed.

• Enumerative geometry— a branch of algebraic geometry concerned with counting the number of solutionsto geometric questions. This is usually done by means of intersection theory.

• Equivariant noncommutative algebraic geometry

• Ergodic Ramsey theory — a branch where problems are motivated by additive combinatorics and solvedusing ergodic theory.

• Ergodic theory— the study of dynamical systems with an invariant measure, and related problems.

• Euclidean geometry

• Euclidean differential geometry— also known as classical differential geometry. See differential geometry.

• Euler calculus

• Experimental mathematics

• Extraordinary cohomology theory

8 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS

• Extremal combinatorics— a branch of combinatorics, it is the study of the possible sizes of a collection offinite objects given certain restrictions.

• Extremal graph theory

1.6 F• Field theory— branch of abstract algebra studying fields.

• Finite geometry

• Finite model theory

• Finsler geometry— a branch of differential geometry whose main object of study is the Finsler manifold (ageneralisation of a Riemannian manifold).

• First order arithmetic

• Fourier analysis

• Fractional calculus— a branch of analysis that studies the possibility of taking real or complex powers ofthe differentiation operator.

• Fractional dynamics— investigates the behaviour of objects and systems that are described by differentiationand integration of fractional orders using methods of fractional calculus.

• Fredholm theory— part of spectral theory studying integral equations.

• Function theory — part of analysis devoted to properties of functions, especially functions of a complexvariable (see complex analysis).

• Functional analysis

• Functional calculus—historically the termwas used synonymously with calculus of variations, but now refersto a branch of functional analysis connected with spectral theory

• Fuzzy arithmetic

• Fuzzy geometry

• Fuzzy Galois theory

• Fuzzy mathematics— a branch of mathematics based on fuzzy set theory and fuzzy logic.

• Fuzzy measure theory

• Fuzzy qualitative trigonometry

• Fuzzy set theory— a form of set theory that studies fuzzy sets, that is sets that have degrees of membership.

• Fuzzy topology

1.7 G• Galois cohomology— an application of homological algebra, it is the study of group cohomology of Galoismodules.

• Galois theory—named after Évariste Galois, it is a branch of abstract algebra providing a connection betweenfield theory and group theory.

• Galois geometry— a branch of finite geometry concerned with algebraic and analytic geometry over a Galoisfield.

• Game theory

1.7. G 9

• Gauge theory

• General topology — also known as point-set topology, it is a branch of topology studying the properties oftopological spaces and structures defined on them. It differs from other branches of topology as the topologicalspaces do not have to be similar to manifolds.

• Generalized trigonometry—developments of trigonometric methods from the application to real numbers ofEuclidean geometry to any geometry or space. This includes spherical trigonometry, hyperbolic trigonometry,gyrotrigonometry, rational trigonometry, universal hyperbolic trigonometry, fuzzy qualitative trigonometry,operator trigonometry and lattice trigonometry.

• Geometric algebra— an alternative approach to classical, computational and relativistic geometry. It showsa natural correspondence between geometric entities and elements of algebra.

• Geometric analysis— a discipline that uses methods from differential geometry to study partial differentialequations as well as the applications to geometry.

• Geometric calculus

• Geometric combinatorics

• Geometric function theory— the study of geometric properties of analytic functions.

• Geometric homology theory

• Geometric invariant theory

• Geometric graph theory

• Geometric group theory

• Geometric measure theory

• Geometric topology— a branch of topology studying manifolds and mappings between them; in particularthe embedding of one manifold into another.

• Geometry—a branch of mathematics concerned with shape and the properties of space. Classically it arose aswhat is now known as solid geometry; this was concerning practical knowledge of length, area and volume. Itwas then put into an axiomatic form by Euclid, giving rise to what is now known as classical Euclidean geometry.The use of coordinates by René Descartes gave rise to Cartesian geometry enabling a more analytical approachto geometric entities. Since thenmany other branches have appeared including projective geometry, differentialgeometry, non-Euclidean geometry, Fractal geometry and algebraic geometry. Geometry also gave rise to themodern discipline of topology.

• Geometry of numbers— initiated by Hermann Minkowski, it is a branch of number theory studying convexbodies and integer vectors.

• Global analysis— the study of differential equations on manifolds and the relationship between differentialequations and topology.

• Global arithmetic dynamics

• Graph theory— a branch of discrete mathematics devoted to the study of graphs. It has many applicationsin physical, biological and social systems.

• Group-character theory— the part of character theory dedicated to the study of characters of group repre-sentations.

• Group representation theory

• Group theory

• Gyrotrigonometry—a form of trigonometry used in gyrovector space for hyperbolic geometry. (An analogyof the vector space in Euclidean geometry.)

10 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS

1.8 H• Hard analysis— see classical analysis

• Harmonic analysis— part of analysis concerned with the representations of functions in terms of waves. Itgeneralizes the notions of Fourier series and Fourier transforms from the Fourier analysis.

• High-dimensional topology

• Higher arithmetic

• Higher category theory

• Higher-dimensional algebra

• Hodge theory

• Holomorphic functional calculus— a branch of functional calculus starting with holomorphic functions.

• Homological algebra— the study of homology in general algebraic settings.

• Homology theory

• Homotopy theory

• Hyperbolic geometry— also known as Lobachevskian geometry or Bolyai-Lobachevskian geometry. It is anon-Euclidean geometry looking at hyperbolic space.

• hyperbolic trigonometry— the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functionsin Euclidean geometry. Other forms include gyrotrigonometry and universal hyperbolic trigonometry.

• Hypercomplex analysis

• Hyperfunction theory

1.9 I• Ideal theory— once the precursor name for what is now known as commutative algebra; it is the theory ofideals in commutative rings.

• Idempotent analysis

• Incidence geometry— the study of relations of incidence between various geometric objects, like curves andlines.

• Inconsistent mathematics— see paraconsistent mathematics.

• Infinitary combinatorics— an expansion of ideas in combinatorics to account for infinite sets.

• Infinitesimal analysis— once a synonym for infinitesimal calculus

• Infinitesimal calculus— see calculus of infinitesimals

• Information geometry

• Integral calculus

• Integral geometry

• Intersection theory— a branch of algebraic geometry and algebraic topology

• Intuitionistic type theory

• Invariant theory— studies how group actions on algebraic varieties affect functions.

• Inversive geometry— the study of invariants preserved by a type of transformation known as inversion

1.10. J 11

• Inversive plane geometry— inversive geometry that is limited to two dimensions• Inversive ring geometry• Itō calculus• Iwasawa theory

1.10 J

1.11 K• K-theory— originated as the study of a ring generated by vector bundles over a topological space or scheme.In algebraic topology it is an extraordinary cohomology theory known as topological K-theory. In algebra andalgebraic geometry it is referred to as algebraic K-theory. In physics, K-theory has appeared in type II stringtheory. (In particular twisted K-theory.)

• K-homology• Kähler geometry — a branch of differential geometry, more specifically a union of Riemannian geometry,complex differential geometry and symplectic geometry. It is the study of Kähler manifolds. (named afterErich Kähler)

• KK-theory• Klein geometry• Knot theory— part of topology dealing with knots• Kummer theory

1.12 L• L-theory• Large deviations theory— part of probability theory studying events of small probability (tail events).• Large sample theory— also known as asymptotic theory• Lattice theory— the study of lattices, being important in order theory and universal algebra• Lattice trigonometry• Lie algebra theory• Lie group theory• Lie sphere geometry• Lie theory• Line geometry• Linear algebra – a branch of algebra studying linear spaces and linear maps. It has applications in fields suchas abstract algebra and functional analysis; it can be represented in analytic geometry and it is generalized inoperator theory and in module theory. Sometimes matrix theory is considered a branch, although linear algebrais restricted to only finite dimensions. Extensions of the methods used belong to multilinear algebra.

• Linear functional analysis• Local algebra— a term sometimes applied to the theory of local rings.• Local arithmetic dynamics— also known as p-adic dynamics or nonarchimedean dynamics.• Local class field theory• Low-dimensional topology

12 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS

1.13 M

• Malliavin calculus

• Mathematical logic

• Mathematical optimization

• Mathematical physics— a part of mathematics that develops mathematical methods motivated by problemsin physics.

• Mathematical sciences— refers to academic disciplines that are mathematical in nature, but are not consid-ered proper subfields of mathematics. Examples include statistics, cryptography, game theory and actuarialscience.

• Matrix algebra

• Matrix calculus

• Matrix theory

• Matroid theory

• Measure theory

• Metric geometry

• Microlocal analysis

• Model theory

• Modern algebra— see abstract algebra

• Modern algebraic geometry— the form of algebraic geometry given by Alexander Grothendieck and Jean-Pierre Serre drawing on sheaf theory.

• Modern invariant theory— the form of invariant theory that analyses the decomposition of representationsinto irreducibles.

• Modular representation theory

• Module theory

• Molecular geometry

• Morse theory— a part of differential topology, it analyzes the topological space of a manifold by studyingdifferentiable functions on that manifold.

• Motivic cohomology

• Multilinear algebra— an extension of linear algebra building upon concepts of p-vectors and multivectorswith Grassmann algebra.

• Multivariable calculus

• Multiplicative number theory—a subfield of analytic number theory that deals with prime numbers, factorizationand divisors.

• Multiple-scale analysis

1.14. N 13

1.14 N• Neutral geometry— see absolute geometry

• Nevanlinna theory— part of complex analysis studying the value distribution of meromorphic functions. Itis named after Rolf Nevanlinna

• Nielsen theory—an area of mathematical research with its origins in fixed point topology, developed by JakobNielsen

• Non-abelian class field theory

• Non-classical analysis

• Non-Euclidean geometry

• Non-standard analysis

• Non-standard calculus

• Nonarchimedean dynamics— also known as p-adic analysis or local arithmetic dynamics

• Noncommutative algebraic geometry — a direction in noncommutative geometry studying the geometricproperties of formal duals of non-commutative algebraic objects.

• Noncommutative geometry

• Noncommutative harmonic analysis— see representation theory

• Noncommutative topology

• Nonlinear analysis

• Nonlinear functional analysis

• Number theory— a branch of pure mathematics primarily devoted to the study of the integers. Originally itwas known as arithmetic or higher arithmetic.

• Numerical analysis

• Numerical geometry

• Numerical linear algebra

1.15 O• Operad theory— a type of abstract algebra concerned with prototypical algebras.

• Operator geometry

• Operator K-theory

• Operator theory— part of functional analysis studying operators.

• Operator trigonometry

• Optimal control theory— a generalization of the calculus of variations.

• Orbifold theory

• Order theory— a branch that investigates the intuitive notion of order using binary relations.

• Ordered geometry— a form of geometry omitting the notion of measurement but featuring the concept ofintermediacy. It is a fundamental geometry forming a common framework for affine geometry, Euclideangeometry, absolute geometry and hyperbolic geometry.

• Oriented elliptic geometry

• Oriented spherical geometry

14 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS

1.16 P• p-adic analysis— a branch of number theory that deals with the analysis of functions of p-adic numbers.

• p-adic dynamics— an application of p-adic analysis looking at p-adic differential equations.

• p-adic Hodge theory

• Parabolic geometry

• Paraconsistent mathematics — sometimes called inconsistent mathematics, it is an attempt to develop theclassical infrastructure of mathematics based on a foundation of paraconsistent logic instead of classical logic.

• Partition theory

• Perturbation theory

• Picard–Vessiot theory

• Plane geometry

• Point-set topology— see general topology

• Pointless topology

• Poisson geometry

• Polyhedral combinatorics— a branch within combinatorics and discrete geometry that studies the problemsof describing convex polytopes.

• Polyhedral geometry

• Possibility theory

• Potential theory

• Precalculus

• Predicative mathematics

• Probability theory

• Probabilistic combinatorics

• Probabilistic graph theory

• Probabilistic number theory

• Projective geometry — a form of geometry that studies geometric properties that are invariant under aprojective transformation.

• Projective differential geometry

• Proof theory

• Pseudo-Riemannian geometry — generalizes Riemannian geometry to the study of pseudo-Riemannianmanifolds.

• Pure mathematics— the part of mathematics that studies entirely abstract concepts.

1.17 Q• Quantum calculus— a form of calculus without the notion of limits. There are 2 forms known as q-calculusand h-calculus

• Quantum geometry— the generalization of concepts of geometry used to describe the physical phenomenaof quantum physics

• Quaternionic analysis

1.18. R 15

1.18 R

• Ramsey theory— the study of the conditions in which order must appear. It is named after Frank P. Ramsey.

• Rational geometry

• Rational trigonometry— a reformulation of trigonometry in terms of spread and quadrance instead of angleand length.

• Real algebra— the study of the part of algebra relevant to real algebraic geometry.

• Real algebraic geometry— the part of algebraic geometry that studies real points of the algebraic varieties.

• Real analysis—abranch ofmathematical analysis; in particular hard analysis, that is the study of real numbersand functions of Real values. It provides a rigorous formulation of the calculus of real numbers in terms ofcontinuity and smoothness, whilst the theory is extended to the complex numbers in complex analysis.

• Real analytic geometry

• Real K-theory

• Recreational mathematics— the area dedicated to mathematical puzzles and mathematical games.

• Recursion theory— see computability theory

• Representation theory— a subfield of abstract algebra; it studies algebraic structures by representing theirelements as linear transformations of vector spaces. It also studies modules over these algebraic structures,providing a way of reducing problems in abstract algebra to problems in linear algebra.

• Representation theory of algebraic groups

• Representation theory of algebras

• Representation theory of diffeomorphism groups

• Representation theory of finite groups

• Representation theory of groups

• Representation theory of Hopf algebras

• Representation theory of Lie algebras

• Representation theory of Lie groups

• Representation theory of the Galilean group

• Representation theory of the Lorentz group

• Representation theory of the Poincaré group

• Representation theory of the symmetric group

• Ribbon theory— a branch of topology studying ribbons.

• Riemannian geometry—a branch of differential geometry that is more specifically, the study of Riemannianmanifolds. It is named after Bernhard Riemann and it featuresmany generalizations of concepts fromEuclideangeometry, analysis and calculus.

• Rough set theory — the a form of set theory based on rough sets.

16 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS

1.19 S

• Scheme theory — the study of schemes introduced by Alexander Grothendieck. It allows the use of sheaftheory to study algebraic varieties and is considered the central part of modern algebraic geometry.

• Secondary calculus

• Semialgebraic geometry—apart of algebraic geometry; more specifically a branch of real algebraic geometrythat studies semialgebraic sets.

• Set-theoretic topology

• Set theory

• Sheaf theory

• Sheaf cohomology

• Sieve theory

• Single operator theory— deals with the properties and classifications of single operators.

• Singularity theory— a branch, notably of geometry; that studies the failure of manifold structure.

• Smooth infinitesimal analysis — a rigorous reformation of infinitesimal calculus employing methods ofcategory theory. As a theory, it is a subset of synthetic differential geometry.

• Solid geometry

• Spatial geometry

• Spectral geometry— a field that concerns the relationships between geometric structures of manifolds andspectra of canonically defined differential operators.

• Spectral graph theory— the study of properties of a graph using methods from matrix theory.

• Spectral theory—part of operator theory extending the concepts of eigenvalues and eigenvectors from linearalgebra and matrix theory.

• Spectral theory of ordinary differential equations— part of spectral theory concerned with the spectrumand eigenfunction expansion associated with linear ordinary differential equations.

• Spectrum continuation analysis— generalizes the concept of a Fourier series to non-periodic functions.

• Spherical geometry— a branch of non-Euclidean geometry, studying the 2-dimensional surface of a sphere.

• Spherical trigonometry— a branch of spherical geometry that studies polygons on the surface of a sphere.Usually the polygons are triangles.

• Statistics—although the termmay refer to the more general study of statistics, the term is used in mathematicsto refer to the mathematical study of statistics and related fields. This includes probability theory.

• Stochastic calculus

• Stochastic calculus of variations

• Stochastic geometry— the study of random patterns of points

• Stratified Morse theory

• Super category theory

• Super linear algebra

• Surgery theory — a part of geometric topology referring to methods used to produce one manifold fromanother (in a controlled way.)

1.20. T 17

• Symbolic computation— also known as algebraic computation and computer algebra. It refers to the tech-niques used to manipulate mathematical expressions and equations in symbolic form as opposed to manipulat-ing them by the numerical quantities represented by them.

• Symbolic dynamics

• Symmetric function theory

• Symplectic geometry — a branch of differential geometry and topology whose main object of study is thesymplectic manifold.

• Symplectic topology

• Synthetic differential geometry— a reformulation of differential geometry in the language of topos theoryand in the context of an intuitionistic logic.

• Synthetic geometry — also known as axiomatic geometry, it is a branch of geometry that uses axioms andlogical arguments to draw conclusions as opposed to analytic and algebraic methods.

• Systolic geometry—abranch of differential geometry studying systolic invariants ofmanifolds and polyhedra.

• Systolic hyperbolic geometry— the study of systoles in hyperbolic geometry.

1.20 T• Tensor analysis—the study of tensors, which play a role in subjects such as differential geometry, mathematicalphysics, algebraic topology, multilinear algebra, homological algebra and representation theory.

• Tensor calculus— an older term for tensor analysis.

• Tensor theory— an alternative name for tensor analysis.

• Theoretical physics—abranch primarily of the science physics that usesmathematical models and abstractionof physics to rationalize and predict phenomena.

• Time-scale calculus

• Topology

• Topological combinatorics— the application of methods from algebraic topology to solve problems in com-binatorics.

• Topological degree theory

• Topological fixed point theory

• Topological graph theory

• Topological K-theory

• Topos theory

• Toric geometry

• Transcendental number theory— a branch of number theory that revolves around the transcendental num-bers.

• Transfinite order theory

• Transformation geometry

• Trigonometry— the study of triangles and the relationships between the length of their sides, and the anglesbetween them. It is essential to many parts of applied mathematics.

• Tropical analysis— see idempotent analysis

18 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS

• Tropical geometry

• Twisted K-theory — a variation on K-theory, spanning abstract algebra, algebraic topology and operatortheory.

• Type theory

1.21 U• Umbral calculus— the study of Sheffer sequences

• Uncertainty theory—a new branch of mathematics based on normality, monotonicity, self-duality, countablesubadditivity, and product measure axioms.

• Unitary representation theory

• Universal algebra— a field studying the formalization of algebraic structures itself.

• Universal hyperbolic trigonometry— an approach to hyperbolic trigonometry based on rational geometry.

1.22 V• Valuation theory

• Variational analysis

• Vector algebra—apart of linear algebra concernedwith the operations of vector addition and scalarmultiplication,although it may also refer to vector operations of vector calculus, including the dot and cross product. In thiscase it can be contrasted with geometric algebra which generalizes into higher dimensions.

• Vector analysis— also known as vector calculus, see vector calculus.

• Vector calculus— a branch of multivariable calculus concerned with differentiation and integration of vectorfields. Primarily it is concerned with 3-dimensional Euclidean space.

1.23 W• Wavelet

• Windowed Fourier transform

• Window function

1.24 X

1.25 Y

1.26 Z

1.27 See also

Glossary of engineering

Chapter 2

Glossary of arithmetic and Diophantinegeometry

This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditionalstudy of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of thetheory is in the form of proposed conjectures, which can be related at various levels of generality.Diophantine geometry in general is the study of algebraic varieties V over fields K that are finitely generated overtheir prime fields—including as of special interest number fields and finite fields—and over local fields. Of those,only the complex numbers are algebraically closed; over any other K the existence of points of V with co-ordinatesin K is something to be proved and studied as an extra topic, even knowing the geometry of V.Arithmetical or arithmetic (algebraic) geometry is a field with a less elementary definition. After the advent of schemetheory it could reasonably be defined as the study ofAlexander Grothendieck's schemes of finite type over the spectrumof the ring of integers Z. This point of view has been very influential for around half a century; it has very widelybeen regarded as fulfilling Leopold Kronecker's ambition to have number theory operate only with rings that arequotients of polynomial rings over the integers (to use the current language of commutative algebra). In fact schemetheory uses all sorts of auxiliary constructions that do not appear at all 'finitistic', so that there is little connection with'constructivist' ideas as such. That scheme theory may not be the last word appears from continuing interest in the'infinite primes’ (the real and complex local fields), which do not come from prime ideals as the p-adic numbers do.Contents :

• Top

• 0–9

• A

• B

• C

• D

• E

• F

• G

• H

• I

• J

• K

19

20 CHAPTER 2. GLOSSARY OF ARITHMETIC AND DIOPHANTINE GEOMETRY

• L

• M

• N

• O

• P

• Q

• R

• S

• T

• U

• V

• W

• X

• Y

• Z

2.1 A

abc conjecture The abc conjecture of Masser and Oesterlé attempts to state as much as possible about repeatedprime factors in an equation a + b = c. For example 3 + 125 = 128 but the prime powers here are exceptional.

Arakelov class group The Arakelov class group is the analogue of the ideal class group or divisor class group forArakelov divisors.[1]

Arakelov divisor An Arakelov divisor (or replete divisor[2]) on a global field is an extension of the concept of divisoror fractional ideal. It is a formal linear combination of places of the field with finite places having integercoefficients and the infinite places having real coefficients.[1][3][4]

Arakelov height The Arakelov height on a projective space over the field of algebraic numbers is a global heightfunction with local contributions coming from Fubini–Study metrics on the Archimedean fields and the usualmetric on the non-Archimedean fields.[5][6]

Arakelov theory Arakelov theory is an approach to arithmetic geometry that explicitly includes the 'infinite primes’.

Arithmetic of abelian varieties See main article arithmetic of abelian varieties

Artin L-functions Artin L-functions are defined for quite general Galois representations. The introduction of étalecohomology in the 1960s meant that Hasse–Weil L-functions could be regarded as Artin L-functions for theGalois representations on l-adic cohomology groups.

2.2. B 21

2.2 B

Bad reduction See good reduction.

Birch and Swinnerton-Dyer conjecture The Birch and Swinnerton-Dyer conjecture on elliptic curves postulates aconnection between the rank of an elliptic curve and the order of pole of its Hasse-Weil L-function. It has beenan important landmark in Diophantine geometry since the mid-1960s, with results such as the Coates–Wilestheorem, Gross–Zagier theorem and Kolyvagin’s theorem.[7]

Bombieri–Lang conjecture Enrico Bombieri, Serge Lang and Paul Vojta and Piotr Blass have conjectured that al-gebraic varieties of general type do not have Zariski dense subsets ofK-rational points, forK a finitely-generatedfield. This circle of ideas includes the understanding of analytic hyperbolicity and the Lang conjectures on that,and the Vojta conjectures. An analytically hyperbolic algebraic variety V over the complex numbers is onesuch that no holomorphic mapping from the whole complex plane to it exists, that is not constant. Examplesinclude compact Riemann surfaces of genus g > 1. Lang conjectured that V is analytically holomorphic if andonly if all subvarieties are of general type.[8]

2.3 C

Canonical height The canonical height on an abelian variety is a height function that is a distinguished quadraticform. See Néron–Tate height.

Chabauty’s method Chabauty’s method, based on p-adic analytic functions, is a special application but capableof proving cases of the Mordell conjecture for curves whose Jacobian’s rank is less than its dimension. Itdeveloped ideas from Thoralf Skolem's method for an algebraic torus. (Other older methods for Diophantineproblems include Runge’s method.)

Coates–Wiles theorem The Coates–Wiles theorem states that an elliptic curve with complex multiplication by animaginary quadratic field of class number 1 and positive rank has L-function with a zero at s=1. This is aspecial case of the Birch and Swinnerton-Dyer conjecture.[9]

Crystalline cohomology Crystalline cohomology is a p-adic cohomology theory in characteristic p, introduced byAlexander Grothendieck to fill the gap left by étale cohomology which is deficient in using mod p coefficientsin this case. It is one of a number of theories deriving in some way from Dwork’s method, and has applicationsoutside purely arithmetical questions.

2.4 D

Diagonal forms Diagonal forms are some of the simplest projective varieties to study from an arithmetic pointof view (including the Fermat varieties). Their local zeta-functions are computed in terms of Jacobi sums.Waring’s problem is the most classical case.

Diophantine dimension The Diophantine dimension of a field is the smallest natural number k, if it exists, such thatthe field of is class Ck: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N > dk. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraicallyclosed fields of dimension 1.[10]

Discriminant of a point The discriminant of a point refers to two related concepts relative to a point P on an alge-braic variety V defined over a number field K: the geometric (logarithmic) discriminant[11] d(P) and the arith-metic discriminant, defined by Vojta.[12] The difference between the two may be compared to the differencebetween the arithmetic genus of a singular curve and the geometric genus of the desingularisation.[12] The arith-metic genus is larger than the geometric genus, and the height of a point may be bounded in terms of the arith-metic genus. Obtaining similar bounds involving the geometric genus would have significant consequences.[12]

22 CHAPTER 2. GLOSSARY OF ARITHMETIC AND DIOPHANTINE GEOMETRY

Dwork’s method Bernard Dwork used distinctive methods of p-adic analysis, p-adic algebraic differential equations,Koszul complexes and other techniques that have not all been absorbed into general theories such as crystallinecohomology. He first proved the rationality of local zeta-functions, the initial advance in the direction of theWeil conjectures.

2.5 EÉtale cohomology The search for a Weil cohomology (q.v.) was at least partially fulfilled in the étale cohomology

theory of Alexander Grothendieck and Michael Artin. It provided a proof of the functional equation for thelocal zeta-functions, and was basic in the formulation of the Tate conjecture (q.v.) and numerous other theories.

2.6 FFaltings height The Faltings height of an elliptic curve or abelian variety defined over a number field is a measure

of its complexity introduced by Faltings in his proof of the Mordell conjecture.[13][14]

Fermat’s last theorem Fermat’s last theorem, the most celebrated conjecture of Diophantine geometry, was provedby Andrew Wiles and Richard Taylor.

Flat cohomology Flat cohomology is, for the school of Grothendieck, one terminal point of development. It hasthe disadvantage of being quite hard to compute with. The reason that the flat topology has been consideredthe 'right' foundational topos for scheme theory goes back to the fact of faithfully-flat descent, the discovery ofGrothendieck that the representable functors are sheaves for it (i.e. a very general gluing axiom holds).

Function field analogy It was realised in the nineteenth century that the ring of integers of a number field hasanalogies with the affine coordinate ring of an algebraic curve or compact Riemann surface, with a point ormore removed corresponding to the 'infinite places’ of a number field. This idea is more precisely encoded inthe theory that global fields should all be treated on the same basis. The idea goes further. Thus elliptic surfacesover the complex numbers, also, have some quite strict analogies with elliptic curves over number fields.

2.7 GGeometric class field theory The extension of class field theory-style results on abelian coverings to varieties of

dimension at least two is often called geometric class field theory.

Good reduction Fundamental to local analysis in arithmetic problems is to reduce modulo all prime numbers p or,more generally, prime ideals. In the typical situation this presents little difficulty for almost all p; for exampledenominators of fractions are tricky, in that reduction modulo a prime in the denominator looks like divisionby zero, but that rules out only finitely many p per fraction. With a little extra sophistication, homogeneouscoordinates allow clearing of denominators bymultiplying by a common scalar. For a given, single point one cando this and not leave a common factor p. However singularity theory enters: a non-singular point may becomea singular point on reduction modulo p, because the Zariski tangent space can become larger when linear termsreduce to 0 (the geometric formulation shows it is not the fault of a single set of coordinates). Good reductionrefers to the reduced variety having the same properties as the original, for example, an algebraic curve havingthe same genus, or a smooth variety remaining smooth. In general there will be a finite set S of primes for agiven variety V, assumed smooth, such that there is otherwise a smooth reduced Vp over Z/pZ. For abelianvarieties, good reduction is connected with ramification in the field of division points by the Néron–Ogg–Shafarevich criterion. The theory is subtle, in the sense that the freedom to change variables to try to improvematters is rather unobvious: see Néron model, potential good reduction, Tate curve, semistable abelian variety,semistable elliptic curve, Serre–Tate theorem.[15]

Grothendieck–Katz conjecture The Grothendieck–Katz p-curvature conjecture applies reduction modulo primesto algebraic differential equations, to derive information on algebraic function solutions. It is an open problemas of 2005. The initial result of this type was Eisenstein’s theorem.

2.8. H 23

2.8 H

Hasse principle The Hasse principle states that solubility for a global field is the same as solubility in all relevantlocal fields. One of the main objectives of Diophantine geometry is to classify cases where the Hasse principleholds. Generally that is for a large number of variables, when the degree of an equation is held fixed. The Hasseprinciple is often associated with the success of the Hardy–Littlewood circle method. When the circle methodworks, it can provide extra, quantitative information such as asymptotic number of solutions. Reducing thenumber of variables makes the circle method harder; therefore failures of the Hasse principle, for example forcubic forms in small numbers of variables (and in particular for elliptic curves as cubic curves) are at a generallevel connected with the limitations of the analytic approach.

Hasse–Weil L-function AHasse–Weil L-function, sometimes called a global L-function, is an Euler product formedfrom local zeta-functions. The properties of such L-functions remain largely in the realm of conjecture, withthe proof of the Taniyama–Shimura conjecture being a breakthrough. The Langlands philosophy is largelycomplementary to the theory of global L-functions.

Height function Aheight function inDiophantine geometry quantifies the size of solutions toDiophantine equations.[16]Classical or naive height is defined in terms of ordinary absolute value on homogeneous coordinates: it is nowusual to take a logarithmic scale, that is, height is proportional to the “algebraic complexity” or number ofbits needed to store a point.[17] Heights were initially developed by André Weil and D. G. Northcott. Inno-vations around 1960 were the Néron–Tate height and the realisation that heights were linked to projectiverepresentations in much the same way that ample line bundles are in pure geometry.

Hilbertian fields A Hilbertian field K is one for which the projective spaces over K are not thin sets in the senseof Jean-Pierre Serre. This is a geometric take on Hilbert’s irreducibility theorem which shows the rationalnumbers are Hilbertian. Results are applied to the inverse Galois problem. Thin sets (the French word ismince) are in some sense analogous to the meagre sets (French maigre) of the Baire category theorem.

2.9 I

Igusa zeta-function An Igusa zeta-function, named for Jun-ichi Igusa, is a generating function counting numbers ofpoints on an algebraic variety modulo high powers pn of a fixed prime number p. General rationality theoremsare now known, drawing on methods of mathematical logic.[18]

Infinite descent Infinite descent was Pierre de Fermat's classical method for Diophantine equations. It becameone half of the standard proof of the Mordell–Weil theorem, with the other being an argument with heightfunctions (q.v.). Descent is something like division by two in a group of principal homogeneous spaces (oftencalled 'descents’, when written out by equations); in more modern terms in a Galois cohomology group whichis to be proved finite. See Selmer group.

Iwasawa theory Iwasawa theory builds up from the analytic number theory and Stickelberger’s theorem as a theoryof ideal class groups as Galois modules and p-adic L-functions (with roots in Kummer congruence on Bernoullinumbers). In its early days in the late 1960s it was called Iwasawa’s analogue of the Jacobian. The analogywas with the Jacobian variety J of a curve C over a finite field F (qua Picard variety), where the finite fieldhas roots of unity added to make finite field extensions F′ The local zeta-function (q.v.) of C can be recoveredfrom the points J(F′) as Galois module. In the same way, Iwasawa added pn-power roots of unity for fixed pand with n→∞, for his analogue, to a number field K, and considered the inverse limit of class groups, findinga p-adic L-function earlier introduced by Kubota and Leopoldt.

2.10 K

K-theory Algebraic K-theory is on one hand a quite general theory with an abstract algebra flavour, and, on theother hand, implicated in some formulations of arithmetic conjectures. See for example Birch–Tate conjecture,Lichtenbaum conjecture.

24 CHAPTER 2. GLOSSARY OF ARITHMETIC AND DIOPHANTINE GEOMETRY

2.11 LLinear torus A linear torus is a geometrically irreducible Zariski-closed subgroup of an affine torus (product of

multiplicative groups).[19]

Local zeta-function A local zeta-function is a generating function for the number of points on an algebraic varietyVover a finite field F, over the finite field extensions of F. According to theWeil conjectures (q.v.) these functions,for non-singular varieties, exhibit properties closely analogous to the Riemann zeta-function, including theRiemann hypothesis.

2.12 MManin–Mumford conjecture TheManin–Mumford conjecture, now proved byMichel Raynaud, states that a curve

C in its Jacobian variety J can only contain a finite number of points that are of finite order in J, unless C =J.[20][21]

Mordell conjecture The Mordell conjecture is now the Faltings theorem, and states that a curve of genus at leasttwo has only finitely many rational points. The Uniformity conjecture states that there should be a uniformbound on the number of such points, depending only on the genus and the field of definition.

Mordell–Lang conjecture The Mordell–Lang conjecture is a collection of conjectures of Serge Lang unifying theMordell conjecture and Manin–Mumford conjecture in an abelian variety or semi-abelian variety.[22][23]

Mordell–Weil theorem The Mordell–Weil theorem is a foundational result stating that for an abelian variety A overa number field K the group A(K) is a finitely-generated abelian group. This was proved initially for numberfields K, but extends to all finitely-generated fields.

Mordellic variety A Mordellic variety is an algebraic variety which has only finitely many points in any finitelygenerated field.[24]

2.13 NNaive height

The naive or classical height of a vector of rational numbers is the maximum absolute value of the vectorof coprime integers obtained by multiplying through by a lowest common denominator. This may beused to define height on a point in projective space over Q, or of a polynomial, regarded as a vector ofcoefficients, or of an algebraic number, from the height of its minimal polynomial.[25]

Néron symbol The Néron symbol is a bimultiplicative pairing between divisors and algebraic cycles on an Abelianvariety used in Néron’s formulation of the Néron–Tate height as a sum of local contributions.[26][27][28] Theglobal Néron symbol, which is the sum of the local symbols, is just the negative of the height pairing.[29]

Néron–Tate height

The Néron–Tate height (also often referred to as the canonical height) on an abelian variety A is aheight function (q.v.) that is essentially intrinsic, and an exact quadratic form, rather than approximatelyquadratic with respect to the addition on A as provided by the general theory of heights. It can be definedfrom a general height by a limiting process; there are also formulae, in the sense that it is a sum of localcontributions.[29]

Nevanlinna invariant The Nevanlinna invariant of an ample divisor D on a normal projective variety X is a realnumber which describes the rate of growth of the number of rational points on the variety with respect to theembedding defined by the divisor.[30] It has similar formal properties to the abscissa of convergence of theheight zeta function and it is conjectured that they are essentially the same.[31]

2.14. O 25

2.14 O

Ordinary reduction AnAbelian varietyA of dimension d has ordinary reduction at a prime p if it has good reductionat p and in addition the p-torsion has rank d.[32]

2.15 Q

Quasi-algebraic closure The topic of quasi-algebraic closure, i.e. solubility guaranteed by a number of variablespolynomial in the degree of an equation, grew out of studies of the Brauer group and the Chevalley–Warningtheorem. It stalled in the face of counterexamples; but see Ax–Kochen theorem from mathematical logic.

2.16 R

Reduction modulo a prime number or ideal See good reduction.

Replete ideal A replete ideal in a number field K is a formal product of a fractional ideal of K and a vector of positivereal numbers with components indexed by the infinite places of K.[33] A replete divisor is an Arakelov divisor.[2]

2.17 S

Sato–Tate conjecture The Sato–Tate conjecture is a conjectural result on the distribution of Frobenius elementsin the Tate modules of the elliptic curves over finite fields obtained from reducing a given elliptic curve overthe rationals. It is due to Mikio Sato and John Tate (independently, and around 1960, published somewhatlater)[34] and is by now supported by very substantial evidence. It is a prototype for Galois representations ingeneral.

Skolem’s method See Chabauty’s method.

Special set The special set in an algebraic variety is the subset in which one might expect to find many rationalpoints. The precise definition varies according to context. One definition is the Zariski closure of the unionof images of algebraic groups under non-trivial rational maps; alternatively one may take images of abelianvarieities;[35] another definition is the union of all subvarieties that are not of general type.[8] For abelian varietiesthe definition would be the union of all translates of proper abelian subvarieties.[36] For a complex variety, theholomorphic special set is the Zariski closure of the images of all non-constant holomorphic maps from C.Lang conjectured that the analytic and algebraic special sets are equal.[37]

Subspace theorem Schmidt’s subspace theorem shows that points of small height in projective space lie in a finitenumber of hyperplanes. A quantitative form of the theorem, in which the number of subspaces containing allsolutions, was also obtained by Schmidt, and the theorem was generalised by Schlickewei (1977) to allow moregeneral absolute values on number fields. The theorem may be used to obtain results on Diophantine equationssuch as Siegel’s theorem on integral points and solution of the S-unit equation.[38]

2.18 T

Tamagawa numbers The direct Tamagawa number definition works well only for linear algebraic groups. Therethe Weil conjecture on Tamagawa numbers was eventually proved. For abelian varieties, and in particular theBirch–Swinnerton-Dyer conjecture (q.v.), the Tamagawa number approach to a local-global principle fails ona direct attempt, though it has had heuristic value over many years. Now a sophisticated equivariant Tamagawanumber conjecture is a major research problem.

26 CHAPTER 2. GLOSSARY OF ARITHMETIC AND DIOPHANTINE GEOMETRY

Tate conjecture The Tate conjecture (John Tate, 1963) provided an analogue to the Hodge conjecture, also onalgebraic cycles, but well within arithmetic geometry. It also gave, for elliptic surfaces, an analogue of theBirch–Swinnerton-Dyer conjecture (q.v.), leading quickly to a clarification of the latter and a recognition of itsimportance.

Tate curve The Tate curve is a particular elliptic curve over the p-adic numbers introduced by John Tate to studybad reduction (see good reduction).

Tsen rank The Tsen rank of a field, named for C. C. Tsen who introduced their study in 1936,[39] is the smallestnatural number i, if it exists, such that the field is of class Ti: that is, such that any system of polynomials withno constant term of degree dj in n variables has a non-trivial zero whenever n > ∑ dji. Algebraically closedfields are of Tsen rank zero. The Tsen rank is greater or equal to the Diophantine dimension but it is not knownif they are equal except in the case of rank zero.[40]

2.19 U

Uniformity conjecture The unformity conjecture states that for any number field K and g > 2, there is a uniformbound B(g,K) on the number of K-rational points on any curve of genus g. The conjecture would follow fromthe Bombieri–Lang conjecture.[41]

Unlikely intersection An unlikely intersection is an algebraic subgroup intersecting a subvariety of a torus or abelianvariety in a set of unusually large dimension, such as is involved in the Mordell-Lang conjecture.[42]

2.20 V

Vojta conjecture The Vojta conjecture is a complex of conjectures by Paul Vojta, making analogies betweenDiophantine approximation and Nevanlinna theory.

2.21 W

Weights The yoga of weights is a formulation by Alexander Grothendieck of analogies between Hodge theory andl-adic cohomology.[43]

Weil cohomology The initial idea, later somewhat modified, for proving the Weil conjectures (q.v.), was to con-struct a cohomology theory applying to algebraic varieties over finite fields that would both be as good assingular homology at detecting topological structure, and have Frobenius mappings acting in such a way thatthe Lefschetz fixed-point theorem could be applied to the counting in local zeta-functions. For later history seemotive (algebraic geometry), motivic cohomology.

Weil conjectures TheWeil conjectures were three highly-influential conjectures of AndréWeil, made public around1949, on local zeta-functions. The proof was completed in 1973. Those being proved, there remain extensionsof the Chevalley–Warning theorem congruence, which comes from an elementary method, and improvementsof Weil bounds, e.g. better estimates for curves of the number of points than come from Weil’s basic theoremof 1940. The latter turn out to be of interest for Goppa codes.

Weil distributions on algebraic varieties André Weil proposed a theory in the 1920s and 1930s on prime idealdecomposition of algebraic numbers in co-ordinates of points on algebraic varieties. It has remained somewhatunder-developed.

Weil function A Weil function on an algebraic variety is a real-valued function defined off some Cartier divisorwhich generalises the concept of Green’s function in Arakelov theory.[44] They are used in the construction ofthe local components of the Néron–Tate height.[45]

2.21. W 27

Weil height machine

TheWeil height machine is an effective procedure for assigning a height function to any divisor on smoothprojective variety over a number field (or to Cartier divisors on non-smooth varieties).[46]

Contents :

• Top

• 0–9

• A

• B

• C

• D

• E

• F

• G

• H

• I

• J

• K

• L

• M

• N

• O

• P

• Q

• R

• S

• T

• U

• V

• W

• X

• Y

• Z

28 CHAPTER 2. GLOSSARY OF ARITHMETIC AND DIOPHANTINE GEOMETRY

2.22 See also

• Arithmetic topology

• Arithmetic dynamics

2.23 References[1] Schoof, René (2008). “Computing Arakelov class groups”. In Buhler, J.P.; P., Stevenhagen. Algorithmic Number Theory:

Lattices, Number Fields, Curves and Cryptography. MSRI Publications 44. Cambridge University Press. pp. 447–495.ISBN 978-0-521-20833-8. MR 2467554. Zbl 1188.11076.

[2] Neukirch (1999) p.189

[3] Lang (1988) pp.74–75

[4] van der Geer, G.; Schoof, R. (2000). “Effectivity of Arakelov divisors and the theta divisor of a number field”. SelectaMathematica, New Series 6 (4): 377–398. arXiv:math/9802121. doi:10.1007/PL00001393. Zbl 1030.11063.

[5] Bombieri & Gubler (2006) pp.66–67

[6] Lang (1988) pp.156–157

[7] Lang (1997) pp.91–96

[8] Hindry & Silverman (2000) p.479

[9] Coates, J.; Wiles, A. (1977). “On the conjecture of Birch and Swinnerton-Dyer”. Inventiones Mathematicae 39 (3): 223–251. doi:10.1007/BF01402975. Zbl 0359.14009.

[10] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathe-matischen Wissenschaften 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 3-540-37888-X.

[11] Lang (1997) p.146

[12] Lang (1997) p.171

[13] Faltings, Gerd (1983). “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”. Inventiones Mathematicae 73 (3):349–366. doi:10.1007/BF01388432.

[14] Cornell, Gary; Silverman, JosephH. (1986). Arithmetic geometry. NewYork: Springer. ISBN 0-387-96311-1. →Containsan English translation of Faltings (1983)

[15] Serre, Jean-Pierre; Tate, John (11968). “Good reduction of abelian varieties”. The Annals of Mathematics. Second 88 (3):492–517. doi:10.2307/1970722. JSTOR 1970722. Zbl 0172.46101. Check date values in: |date= (help)

[16] Lang (1997) pp.43–67

[17] Bombieri & Gubler (2006) pp.15–21

[18] Igusa, Jun-Ichi (1974). “Complex powers and asymptotic expansions. I. Functions of certain types”. Journal für die reineund angewandte Mathematik 1974 (268–269): 110–130. doi:10.1515/crll.1974.268-269.110. Zbl 0287.43007.

[19] Bombieri & Gubler (2006) pp.82–93

[20] Raynaud, Michel (1983). “Sous-variétés d'une variété abélienne et points de torsion”. In Artin, Michael; Tate, John.Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic.Progress in Mathematics (in French) 35. Birkhauser-Boston. pp. 327–352. Zbl 0581.14031.

[21] Roessler, Damian (2005). “A note on the Manin-Mumford conjecture”. In van der Geer, Gerard; Moonen, Ben; Schoof,René. Number fields and function fields — two parallel worlds. Progress in Mathematics 239. Birkhäuser. pp. 311–318.ISBN 0-8176-4397-4. Zbl 1098.14030.

[22] Marcja, Annalisa; Toffalori, Carlo (2003). A Guide to Classical and Modern Model Theory. Trends in Logic 19. Springer-Verlag. pp. 305–306. ISBN 1402013302.

[23] 2 page exposition of the Mordell-Lang conjecture by B. Mazur, 3 Nov. 2005

2.24. FURTHER READING 29

[24] Lang (1997) p.15

[25] Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs9. Cambridge University Press. p. 3. ISBN 978-0-521-88268-2. Zbl 1145.11004.

[26] Bombieri & Gubler (2006) pp.301–314

[27] Lang (1988) pp.66–69

[28] Lang (1997) p.212

[29] Lang (1988) p.77

[30] Hindry & Silverman (2000) p.488

[31] Batyrev, V.V.; Manin, Yu.I. (1990). “On the number of rational points of bounded height on algebraic varieties”. Math.Ann. 286: 27–43. doi:10.1007/bf01453564. Zbl 0679.14008.

[32] Lang (1997) pp.161–162

[33] Neukirch (1999) p.185

[34] It is mentioned in J. Tate, Algebraic cycles and poles of zeta functions in the volume (O. F. G. Schilling, editor), ArithmeticalAlgebraic Geometry, pages 93–110 (1965).

[35] Lang (1997) pp.17–23

[36] Hindry & Silverman (2000) p.480

[37] Lang (1997) p.179

[38] Bombieri & Gubler (2006) pp.176–230

[39] Tsen, C. (1936). “Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper”. J. Chinese Math.Soc. 171: 81–92. Zbl 0015.38803.

[40] Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. pp. 109–126.ISBN 978-0-387-72487-4.

[41] Caporaso, Lucia; Harris, Joe; Mazur, Barry (1997). “Uniformity of rational points”. Journal of the American MathematicalSociety 10 (1): 1–35. doi:10.2307/2152901. Zbl 0872.14017.

[42] Zannier, Umberto (2012). Some Problems of Unlikely Intersections in Arithmetic and Geometry. Annals of MathematicsStudies 181. Princeton University Press. ISBN 978-0-691-15371-1.

[43] Pierre Deligne, Poids dans la cohomologie des variétés algébriques, Actes ICM, Vancouver, 1974, 79–85.

[44] Lang (1988) pp.1–9

[45] Lang (1997) pp.164,212

[46] Hindry & Silverman (2000) 184–185

• Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs4. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034.

• Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts inMathematics 201. ISBN 0-387-98981-1. Zbl 0948.11023.

• Lang, Serge (1988). Introduction to Arakelov theory. New York: Springer-Verlag. ISBN 0-387-96793-1. MR0969124. Zbl 0667.14001.

• Lang, Serge (1997). Survey ofDiophantine Geometry. Springer-Verlag. ISBN3-540-61223-8. Zbl 0869.11051.

• Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften 322.Springer-Verlag. ISBN 978-3-540-65399-8. Zbl 0956.11021.

2.24 Further reading• Dino Lorenzini (1996), An invitation to arithmetic geometry, AMS Bookstore, ISBN 978-0-8218-0267-0

Chapter 3

Glossary of classical algebraic geometry

The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction ofthe general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of thecentury, and later formalized by André Weil, Serre and Grothendieck. Much of the classical terminology, mainlybased on case study, was simply abandoned, with the result that books and papers written before this time can behard to read. This article lists some of this classical terminology, and describes some of the changes in conventions.Dolgachev (2012) translates many of the classical terms in algebraic geometry into scheme-theoretic terminology.Other books defining some of the classical terminology include Baker (1922), Coolidge (1931), Coxeter (1969),Hudson (1990), Salmon (1879), Semple & Roth (1949).Contents :

• Conventions

• !$@

• A

• B

• C

• D

• E

• F

• G

• H

• I

• J

• K

• L

• M

• N

• O

• P

• Q

30

3.1. CONVENTIONS 31

• R

• S

• T

• U

• V

• W

• XYZ

•

•

• See also

• References

3.1 Conventions

On the other hand, while most of the material treated in the book exists in classical treatises in algebraic geometry,their somewhat archaic terminology and what is by now completely forgotten background knowledge makes thesebooks useful to but a handful of experts in the classical literature.(Dolgachev 2012, p.iii–iv)

The change in terminology from around 1948 to 1960 is not the only difficulty in understanding classical algebraicgeometry. There was also a lot of background knowledge and assumptions, much of which has now changed. Thissection lists some of these changes.

• In classical algebraic geometry, adjectives were often used as nouns: for example, “quartic” could also be shortfor “quartic curve” or “quartic surface”.

• In classical algebraic geometry, all curves, surfaces, varieties, and so on came with fixed embeddings intoprojective space, whereas in scheme theory they are more often considered as abstract varieties. For example,a Veronese surface was not a just copy of the projective plane, but a copy of the projective plane together withan embedding into projective 5-space.

• Varieties were often considered only up to birational isomorphism, whereas in scheme theory they are usuallyconsidered up to biregular isomorphism. (Semple & Roth 1949, p.20–21)

• Until circa 1950, many of the proofs in classical algebraic geometry were incomplete (or occasionally justwrong). In particular authors often did not bother to check degenerate cases.

• Words (such as azygetic or bifid) were sometimes formed fromLatin orGreek roots without further explanation,assuming that readers would use their classical education to figure out the meaning.

...we refer to a certain degree of informality of language, sacrificing precision to brevity, ..., and which has longcharacterized most geometrical writing. ...[The meaning] depends always on the context and is invariably assumedto be capable of unambiguous interpretation by the reader.(Semple & Roth 1949, p.iii)

32 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

• Definitions in classical algebraic geometry were often somewhat vague, and it is futile to try to find the precisemeaning of some of the older terms because many of them never had a precise meaning. In practice this didnot matter much when the terms were only used to describe particular examples, as in these cases their meaningwas usually clear: for example, it was obvious what the 16 tropes of a Kummer surface were, even if “trope”was not precisely defined in general.

• Algebraic geometry was often implicitly done over the complex numbers (or sometimes the real numbers).

• Readers were often assumed to know classical (or synthetic) projective geometry, and in particular to have athorough knowledge of conics, and authors would use terminology from this area without further explanation.

• Several terms, such as “Abelian group”, “complete”, “complex”, “flat”, “harmonic”, “homology”, “monoid”,“normal”, “pole”, “regular”, now have meanings that are unrelated to their original meanings. Other terms,such as “circle”, have their meanings tacitly changed to work in complex projective space; for example, a circlein complex algebraic geometry is a conic passing through the circular points at infinity and has underlyingtopological space a 2-sphere rather than a 1-sphere.

• Sometimes capital letters are tacitly understood to stand for points, and small letters for lines or curves.

3.2 Symbols[1], [2], . . . , [n] Projective space of dimension 1, 2, ..., n. This notation was introduced by Schubert (1886)

∞¹, ∞², ... A family of dimension 1, 2, ...

{1}, {2}, ...,{n} A family or variety of dimension 1, 2, ..., n. (Semple & Roth 1949, p.288)

3.3 AAbelian group 1. An archaic name for the symplectic group.

2. A commutative group.

aberrancy The deviation of a curve from circular form. See Salmon (1879, p. 356)

absolute 1. A fixed choice of something in projective space, used to construct some other geometry from projectivegeometry. For example, choosing a plane, called the absolute plane, of projective space can be used to makeits complement into a copy of affine space. Choosing a suitable conic or polarity, called the Cayley absolute,absolute conic or absolute polarity, in the absolute plane provides the means to put a metric on affine spaceso that it becomes a metric space.

2. Absolute geometry is roughly Euclidean geometry without the parallel postulate.

accidental An accidental (or improper) double point of a surface in 4-dimensional projective space is a double pointwith two distinct tangent planes. (Baker 1933, vol 6, p. 157)

acnode An acnode is an isolated point of a real curve. See Salmon (1879, p.23)

adjoint If C is a curve, an adjoint of C is a curve such that any point of C of multiplicity r has multiplicity at leastr–1 on the adjoint. Sometimes the multiple points of C are required to be ordinary, and if theis condition isnot satisfied the term “sub-adjoint” is used. (Semple & Roth 1949, p.55, 231)

affine 1. Affine space is roughly a vector space where one has forgotten which point is the origin

2. An affine variety is a variety in affine space

affinity An automorphism of affine space

aggregate A set.

3.4. B 33

ambient An ambient variety is a large variety containing all the points, curves, divisors, and so on that one is inter-ested in.

anharmonic ratio Cross-ratio

antipoint One of a pair of points constructed from two foci of a curve. See Salmon (1879, p.119)

apparent An apparent singularity is a singularity of a projection of a variety into a hyperplane. They are so calledbecause they appear to be singularities to an observer at the point being projected from. (Semple & Roth 1949,p.55, 231)

apolar Orthogonal under the polar pairing between the symmetric algebra of a vector space and its dual

arithmetic genus The arithmetic genus of a variety is a variation of the Euler characteristic of the trivial line bundle;see Hodge number.

Aronhold set One of the 288 sets of 7 of the 28 bitangents of a quartic curve corresponding to the 7 odd thetacharacteristics of a normal set.

associated 1. An associated curve is the image of a projective curve in a Grassmannian, given by taking the tangentlines, or osculating planes, and so on.

axis, axial A special line or linear subspace associated with some family of geometric objects. For example, a speciallinear complex in 4-dimensional space consists of all lines meeting a given plane, that is called the axial planeof the complex. (Semple & Roth 1949, p.274) Similar to directrix.

azygetic Unpaired. Opposite of syzygetic, meaning paired. Example: azygetic triad, azygetic tetrad, azygetic set.

3.4 B

base 1. A base point is a point common to all members of a family

2. The base number ρ is the rank of the Neron–Severi group.

bicircular Having nodes at the two circular points at infinity, as in bicircular curve. See Salmon (1879, p.231)

bicorn A bicorn is a curve with two cusps

bicuspidal Having two cusps

bidegree A pair of integers giving the degrees of a bihomogeneous polynomial in two sets of variables

bielliptic 1. A bielliptic curve is a branched double cover of an elliptic curve.

2. A bielliptic surface is the same as a hyperelliptic surface.

bifid 1. Split into two equal parts

2. A bifid map is an element of the vector space of dimension 2g over the field with 2 elements, consisting of the2g+1-dimensional space of even-cardinality subsets of a set S of 2+2g elements, modulo the 1-dimensionalspace {0,S}. (Dolgachev 2012, p.215)

3. A bifid substitution is a permutation of the 28 bitangents of a quartic curve depending on one of the 35decompositions of 8 symbols into two sets of 4 symbols. See Salmon (1879, p.223)

biflecnode Same as fleflecnode. See (Salmon 1879, p.210)

bigenus The second plurigenus P2 of a surface.

bihomogeneous Homogeneous in each of two sets of variables, as in bihomogeneous form.

binary Depending on two variables, as in binary form

binodal Having two nodes

34 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

binode A double point of a surface whose tangent cone consists of two different planes. See unode. (Semple & Roth1949, p.424)

bipartite Having two connected components. See Salmon (1879, p.165)

bipunctual 1. Having two points

2. For a bipunctual conic with respect to 3 points see (Baker 1922, vol 2, p. 123).

birational Two varieties are birational if they are isomorphic off lower-dimensional subsets

2. A birational map is a rational map with rational “inverse”

biregular 1. A biregular map is a regular map with regular inverse

2. Two varieties are biregular if there is a biregular map from one to the other, in other words if they are isomorphicas abstract varieties.

biscribed Both circumscribed and inscribed, or in other words having vertices that lie on a curve and sides that aretangent to the curve, as in biscribed triangle. (Dolgachev 2012)

bitangent A bitangent is a line that is tangent to a curve at two points. See Salmon (1879, p. 328)

bitangential Meeting a curve at the tangency points of its bitangents

Brianchon hexagon A non-planar hexagon whose three diagonals meet. (Baker 1922, p.47)

3.5 C

canonical 1. The canonical series is the linear series of the canonical line bundle

2. The canonical bundle is the line bundle of differential forms of highest degree.

3. The canonical map or canonical embedding is the map to the projective space of the sections of the canonicalbundle

4. A canonical curve (or variety) is the image of a curve (or variety) under the canonical map

5. The canonical class is the divisor class of a canonical divisor

6. A canonical divisor is a divisor of a section of the canonical line bundle.

catalecticant A catalecticant is an invariant of a binary form of degree 2n that vanishes when the form is a sum ofpowers of n linear forms.

caustic A caustic is the envelope of light rays from a point reflected in a curve

Cayley

Cayleyan Named after Arthur Cayley

1.Main article: CayleyanSee Salmon (1879)

2. A Cayley octad is a set of 8 points in projective space given by the intersection of three quadrics. (Dolgachev2012, 6.3.1)

3. The Cayley lines or Cayley–Salmon lines are the 20 lines passing through 3 Kirkman points.

4. A Cayley absolute is a conic or quadric used to define a metric.

center

centre 1. A special point associated with some geometric object

3.5. C 35

2. The center of a perspectivity

3. The center of an isologue

character

characteristic 1. An integer associated with a projective variety, such as its degree, rank, order, class, type. (Semple&Roth 1949, p.189) In particular thePlücker characteristics of a curve are the order, class, number of nodes,number of bitangents, number of cusps, and number of inflections. (Coolidge 1931, p.99)

2. A characteristic exponent is an exponent of a power series with non-negative coefficient, that is not divisible bythe highest common factor of preceding exponents with non-zero coefficients. (Coolidge 1931, p.220)

3. The characteristic series of a linear system of divisors on a surface is the linear system of 0-cycles on one of thedivisors given by its intersections with the other divisors.

chord A line joining two points of a variety

chordal variety A chordal variety is the union of the chords and tangent spaces of a projective variety

circle A plane conic passing through the circular points at infinity. For real projective geometry this is much thesame as a circle in the usual sense, but for complex projective geometry it is different: for example, cicles haveunderlying topological spaces given by a 2-sphere rather than a 1-sphere.

circuit A component of a real algebraic curve. A circuit is called even or odd depending on whether it has an evenor odd number of intersections with a generic line. (Coolidge 1931, p. 50)

circular 1. A circular point is one of the two points at infinity (1: i: 0), (1: −i: 0) through which all circles pass

2. A circular algebraic curve is a curve passing through the two circular points at infinity. See also bicircular.

circumscribed 1. Having edges tangent to some curve, as in circumscribed quadrilateral.

2. Passing through the vertices of something, as in circumscribed circle.

cissoid A cissoid is the curve generated from two curves and a point. See Salmon (1879)

class 1. The class of a plane curve is the number of proper tangents passing through a generic point of the plane.(Semple & Roth 1949, p.28)

2. The class of a space curve is the number of osculating planes passing through a generic point of space. (Semple& Roth 1949, p.85)

3. The class of a surface in rdimensional projective space is the number of tangent planes meeting a generic codi-mension 2 subspace in a line. (Semple & Roth 1949, p.28)

4. The degree of a contravariant or concomitant in the covariant variables.

coaxal

coaxial A pencil of circles is called coaxal if their centers all lie on a line (called the axis).

A family of plane circles all passing through the same two points (other than the circular points at infinity). (Baker1922, vol 2, p. 66)

coincidence 1. A coincidence quadric is a quadric associated to a correlation, given by the locus of points lying inthe corresponding hyperplane. (Semple & Roth 1949, p.8)

2. A fixed point of a correspondence, in other words a point of a variety corresponding to itself under a correspon-dence (Coolidge 1931, p. 126)

collinear On the same line

collineation A collineation is an isomorphism from one projective space to another, often to itself (Semple & Roth1949, p.6) See correlation.

complete 1. A linear series of divisors is called complete if it is not contained in a larger linear series.(Semple &Roth 1949, p.351)

36 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

2. A scheme is called complete if the map to a point is proper

3. A complete quadrangle is 4 points and the 6 lines joining pairs

4. A complete quadrilateral is 4 lines meeting in pairs in 6 points

complex 1. (Noun.) A line complex, a family of lines of codimension 1 in the family of all lines in some projectivespace, in particular a 3-dimensional family of lines in 3-dimensional projective space. (Semple & Roth 1949,p.236) See congruence.

2. (Adjective.) Related to the complex numbers.

3. The (line) complex group is an old name for the symplectic group.

composite Reducible (meaning having more than one irreducible component).

conchoid A conchoid is the curve given by the cissoid of a circle and another curve. See Salmon (1879)

concomitant A (mixed) concomitant is an invariant homogeneous polynomial in the coefficients of a form, a covari-ant variable, and a contravariant variable. In other words it is a (tri)homogeneous polynomial on SV⊕V⊕V*for some vector space V, where SV is some symmetric power of V and V* its dual, that is invariant under thespecial linear group of V. In practice V often has dimension 2. The degree, class, and order of a concomitantare its degrees in the three types of variable. Concomitants are generalizations of covariants, contravariants,and invariants.

concurrent Meeting at a point

cone 1. The union of the lines joining an algebraic set with a linear algebraic set. Called a point-cone, line-cone, ...if the linear set is a point, line, ...(Semple & Roth 1949, p.18)

2. A subset of a vector space closed under multiplication by scalars.

configuration A configuration is a finite set of points and lines (and sometimes planes), generally with equal numbersof points per line and equal numbers of lines per point.

confocal Having the same foci

congruence A family of lines in projective space such that there are a nonzero finite number of lines through ageneric point (Semple & Roth 1949, p.238, 288). See complex.

conic A conic is a degree 2 curve. Short for “conic section”, the intersection of a cone with a plane.

conjugate 1. A conjugate point is an acnode. (Salmon 1879, p.23)

2. A conjugate point is a point lying on the hyperplane corresponding to another point under a polarity.

3. A conjugate line is a line containing the point corresponding to another line under a polarity (or plane conic).(Baker & 1922 vol2, p.26)

4. For harmonic conjugate see harmonic.

connex A correspondence between a projective space and its dual.

consecutive Infinitesimally near. For example, a tangent line to a curve is a line through two consecutive points ofthe curve, and a focal point is the intersection of the normals of two consecutive points.

contravariant 1. A bihomogeneous polynomial in dual variables of x, y, ... and the coefficients of some homo-geneous form in x, y,... that is invariant under some group of linear transformations. In other words it is abihomogeneous polynomial on SV⊕V for some vector space V, where SV is some symmetric power of V andV* its dual, that is invariant under the special linear group of V. In practice V often has dimension at least3, because when it has dimension 2 these are more or less the same as covariants. The degree and class of acontravariant are its degrees in the two types of variable. Contravariants generalize invariants and are specialcases of concomitants, and are in some sense dual to covariants.

coplanar In the same plane

3.6. D 37

correlation An isomorphism from a projective space to the dual of a projective space, often to the dual of itself. Acorrelation on the projective space of a vector space is essentially the same as a nonsingular bilinear form onthe vector space, up to multiplication by constants. (Semple & Roth 1949, p.7)

coresidual See Salmon (1879, p.131)

correspondence A correspondence from X to Y is an algebraic subset of X×Y

cosingular Having the same singularities

couple An ordered pair

covariant 1. A bihomogeneous polynomial in x, y, ... and the coefficients of some homogeneous form in x, y,... thatis invariant under some group of linear transformations. In other words it is a bihomogeneous polynomial onSV⊕V* for some vector space V, where SV is some symmetric power of V and V* its dual, that is invariantunder the special linear group ofV. In practiceV often has dimension 2. The degree and order of a covariant areits degrees in the two types of variable. Covariants generalize invariants and are special cases of concomitants,and are in some sense dual to contravariants

2. The variety defined by a covariant. In particular the curve defined by the Hessian or Steinerian covariants of acurve are called covariant curves. (Coolidge 1931, p.151)

Cremona transformation A Cremona transformation is a birational map from a projective space to itself

cross-ratio The cross-ratio is an invariant of 4 points on a projective line.

crunode Crunode is an archaic term for a node, a double point with distinct tangent directions.

cubic Degree 3, especially a degree 3 projective variety

cubo-cubic A cubo-cubic transformation is a Cremona transformation such that the homaloids of the transformationand its inverse all have degree 3. Semple & Roth (1949, p.179)

curve A curve together with an embedding into projective space.

cusp A cusp is a singular point of a curve whose tangent cone is a line

cuspidal edge The locus of the focal points of a family of planes (Semple & Roth 1949, p.85, 87)

cyclide A cyclide is a quartic surface passing doubly through the absolute conic. (Semple & Roth 1949, p.141)

3.6 D

decic

decimic 1. (Adjective) Degree 10

2. (Noun) A degree 10 projective variety

deficiency 1. The deficiency of a linear system is its codimension in the corresponding complete linear system.

2. The deficiency D of a plane curve is an approximation to its genus, equal to the genus when all singular pointsare ordinary, given by (n–1)(n–2)/2 –(a–1)(a–2)/2 – (b–1)(b–2)/2 –..., where n is the degree of the curve anda. b, ... are the multiplicities of its singular points. (Semple & Roth 1949, p.30), (Salmon 1879, p. 28)

degree 1. The number of intersection points of a projective variety with a generic linear subspace of complementarydimension

2. The number of points of a divisor on a curve

Desargues The Desargues figure or configuration is a the configuration of 10 lines and 10 points in Desargues’theorem.

desmic system A desmic system is a configuration of three desmic tetrahedra.

38 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

developable 1. (Noun) A 1-dimensional family of planes in 3-dimensional projective space (Semple & Roth 1949,p.85).

2. (Noun) The envelope of the normals of a curve

3. (Noun) Short for a developable surface, one that can be unrolled to a plane

4. The tangent developable of a curve is the surface consisting of its tangent lines.

5. Flat, as in developable surface

director The director circle of a conic is the locus of points where two orthogonal tangent lines to the conic meet.More generally the director conic of a conic in regard to two points is defined in a similar way. (Baker 1922,vol 2, p. 26)

directrix A straight line, or more generally a projective space, associated with some geometric configuration, suchas the directrix of a conic section or the directrix of a rational normal scroll

discriminant The invariant (on the vector space of forms of degree d in n variables) which vanishes exactly whenthe corresponding hypersurface in Pn-1 is singular.

double curve A 1-dimensional singularity, usually of a surface, of multiplicity 2

double point 1. A 0-dimensional singularity of multiplicity 2, such as a node.

One of the two points fixed by an involution of a projective line. (Baker & 1922 vol 2, p.3)

double six The Schläfli double six configuration

duad A set of two points

dual 1. The dual of a projective space is the set of hyperplanes, considered as another projective space

2. The dual curve of a plane curve is the set of its tangent lines, considered as a curve in the dual projective plane.

3. A dual number is a number of the form a+εb where ε has square 0. Semple & Roth (1949, p.268)

3.7 EEckardt point An Eckardt point is a point of intersection of 3 lines on a cubic surface.

effective An effective cycle or divisor is one with no negative coefficients

elation A collineation that fixes all points on a line (called its axis) and all lines though a point on the axis (called itscenter).

eleven-point conic The eleven-point conic is a conic containing 11 special points associated to four points and aline. (Baker 1922, vol 2, p. 49)

embedded An embedded variety is one contained in a larger variety, sometimes called the ambient variety.

enneaedro A set of 9 tritangent planes to a cubic surface containing the 27 lines.

envelope A curve tangent to a family of curves. See Salmon (1879, p. 65)

epitrochoid An epitrochoid is the curve traced by a point of a disc rolling along another disc. Salmon (1879)

equiaffine

equiaffinity An equiaffinity is an equiaffine transformation, meaning an affine transformation preserving area.

equianharmonic 1. Four points whose cross ratio (or anharmonic ratio) is a cube root of 1

2. An equianharmonic cubic is a cubic curve with j-invariant 0

equivalence In intersection theory, a positive-dimensional variety sometimes behaves formally as if it were a finitenumber of points; this number is called its equivalence.

3.8. F 39

evectant A contravariant defined by Sylvester depending on an invariant. See Salmon (1879, p. 184)

evolute An evolute is the envelope of the normal lines of a plane curve. See Salmon (1879, p. 40)

exceptional 1. Corresponding to something of lower dimension under a birational correspondence, as in exceptionalcurve, exceptional divisor

2. An exceptional curve on a surface is one that corresponds to a simple point on another surface under a birationalcorrespondence. It is called an exceptional curve of the first kind if it is transformed into a point of the othersurface, and an exceptional curve of the second kind if it is transformed into a curve of the other surface.

3.8 F

facultative A facultative point is one where a given function is positive. (Salmon 1885, p.243)

flat 1. (Noun) A linear subspace of projective space, such as a point. line, plane, hyperplane.

2. (Adjective) Having curvature zero

3. (Adjective) For the term “flat” in scheme theory see flat module, flat morphism.

flecnode A double point that is also a point of inflexion of one branch. (Cayley 1852). (Salmon 1879, p.210)

fleflecnode A double point that is also a point of inflexion of both branches. (Cayley 1852).

flex Short for point of inflection

focal 1. A focal point, line, plane, ... is the intersection of several consecutive elements of a family of linearsubspaces. (Semple & Roth 1949, p. 85, 252)

2. A focal curve, surface and so on is the locus of the focal points of a family of linear subspaces. (Semple & Roth1949, p.252)

focus A focal point. See Salmon (1879, p. 116), (Semple & Roth 1949, p. 85,251)

foliate singularity See (Semple & Roth 1949, p.422)

form 1. A homogeneous polynomial in several variables. Same as quantic.

2. A differential form.

free intersection An intersection point of two members of a family that is not a base point.

freedom Dimension, as in degrees of freedom. (Semple & Roth 1949, p.26).

fundamental This term seem to be ambiguous and poorly defined: Zariski states: “I can find no clear-cut definitionof a fundamental curve in the literature”.

1. The fundamental set or fundamental locus of a birational correspondence appears to mean (roughly) either theset of points where it is not a bijection or the set of points where it is not defined.

2. A fundamental point, curve, or variety is a point, curve, or variety in the fundamental set of a birational corre-spondence.

3.9 G

grd, γrd A linear or algebraic system of divisors of dimension r and degree d on a curve. The letter g is used forlinear systems, and the letter γ is used for algebraic systems

40 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

generator One of the lines of a ruled surface (Semple & Roth 1949, p.204) or more generally an element of somefamily of linear spaces.

Most particularly we refer to the recurrent use of such adjectives as `general' or `generic', or such phrases as `ingeneral', whose meaning, wherever they are used, depends always on the context and is invariably assumed tobe capable of unambiguous interpretation by the reader.(Semple & Roth 1949, p.iii)

generic 1. Not having some special properties, which are usually not stated explicitly.

2. A generic point is one having coordinates that are algebraically independent over the base field.

3. The generic point of a scheme.

genus 1. The dimension of the space of sections of the canonical bundle, as in the genus of a curve or the geometricgenus of a surface

2. arithmetic genus of a surface

3. plurigenus

geometric genus The geometric genus is the dimension of the space of holomorphic n-forms on an n-dimensionalnon-singular projective variety.

grade The grade of a linear system of divisors on an n-dimensional variety is the number of free intersection pointsof n generic divisors. In particular the grade of a linear series of divisors on a curve is now called the degreeand is the number of points in each divisor (Semple & Roth 1949, p.345), and the grade of a net of curves ona surface is the number of free intersections of two generic curves. (Semple & Roth 1949, p.45) (Semple &Roth 1949, p.159)

Grassmannian A Grassmannian is a variety parameterizing linear subspaces of projective space

group 1. A group or point-group is an archaic term for an effective divisor on a curve. This usage is particularlyconfusing, because some such divisors are called normal, with the result that there are “normal sub-groups”having nothing to do with the normal subgroups of group theory. (Coolidge 1931)

2. A group in the usual sense.

3.10 H

harmonic 1. Two pairs of points on a line are harmonic if their cross ratio is –1. The 4 points are called a harmonicset, and the points of one pair are called harmonic conjugates with respect to the other pair.

2. A harmonic cubic is an elliptic curve with j-invariant 1728, given by a double cover of the projective line branchedat 4 points with cross ratio –1

3. Satisfying some analogue of the Laplace equation, as in harmonic form.

4. The harmonic polar line of an inflection point of a cubic curve is the component of the polar conic other thanthe tangent line. (Dolgachev 2012, 3.1.2)

5. A harmonic net is a set of points on a line containing the harmonic conjugate of any point with respect to anyother two points. (Baker 1922, p.133)

6. For harmonically conjugate conics see (Baker 1922, vol 2, p.122).

Hesse

Hessian Named after Otto Hesse.

1. A Hessian matrix, or a variety associated with it. See Salmon (1879, p.55)

3.10. H 41

2. The Hessian line is a line associated to 3 points A, B, C, of a conic, containing the three points given by theintersections of the tangents at A, B, C with the lines BC, CA, AB.

3. The Hessian point is a point associated to three lines tangent to a conic, whose construction is dual to that of aHessian line

4. The Hessian pair or Hessian duad of three points on a projective line is the pair of points fixed by the projectivetransformations of order 3 permuting the 3 points. More generally the Hessian pair is also defined in a similarway for triples of points of a rational curve, or triples of elements of a pencil.

5. The Hesse configuration is the configuration of inflection points of a plane cubic.

6. The Hesse group is the group of automorphisms of the Hesse configuration, of order 216.

hexad A set of 6 points

homaloid An element of a homaloidal system, in particular the image of a hyperlpane under a Cremona transfor-mation.

homaloidal 1. A homaloidal linear system of divisors is a linear system of grade 1, such as the image of the linearsystem of hyperplanes of projective space under a Cremona transformation. (Semple & Roth 1949, p.45)(Coolidge 1931, p. 442) When the linear system has dimension 2 or 3 it is called a homaloidal net or homa-loidal web.

2. Homaloidal means similar to a flat plane.

homographic 1. Having the same invariants. See Salmon (1879, p.232)

2. A homographic transformation is an automorphism of projective space over a field, in other words an element ofthe projective general linear group. (Salmon 1879, p.283)

homography 1. An isomorphism between projective spaces induced by an isomorphism of vector spaces.

2. An axis of homography is a line associated to two related ranges of a conic. (Baker 1922, vol2, p.16)

homology 1. As in homology group

2. A collineation fixing all lines through a point (the center) and all points through a line (the axis) not containingthe center. See elation. This terminology was introduced by Lie.

3. An automorphism of projective space with a hyperplane of fixed points (called the axis). It is called a harmonichomology if it has order 2, in which case it has an isolated fixed point called its center.

Hurwitz curve

Hurwitz surface A Hurwitz curve is a complex algebraic curve of genus g>0 with the maximum possible number84(g–1) of automorphisms.

hyperbolism Essentially a blow-up of a curve at a point. See Salmon (1879, p.175)

hypercusp A singularity of a curve of some multiplicity r whose tangent cone is a single line meeting the curve withorder r+1. (Coolidge 1931, p. 18)

hyperelliptic A hyperelliptic curve is a curve with a degree 2 map to the projective line.

hyperflex Same as point of undulation: a point of a curve where the tangent line has contact of order at least 4.

hyperosculating point A point where the tangent space meets with order higher than normal.

hyperplane A linear subspace of projective space of codimension 1. Same as prime.

42 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

3.11 I

index of speciality The dimension of the first cohomology group of the line bundle of a divisor D; often denoted byi or i(D). Semple & Roth (1949, p.381)

infinitely near point A point on a blow up of a variety

inflection

inflexion An inflection is a point where the curvature vanishes, or in other words where the tangent line meets withorder at least 3. Differential geometry uses the slightly stricter condition that the curvature changes sign at thepoint. See Salmon (1879, p. 32)

inpolar quadric See (Baker 1922, volume III, p. 52, 88)

inscribed 1. Having vertices on a curve, as in inscribed figure.

2. Tangent to some lines, as in inscribed circle.

integral An integral is (more or less) what is now called a closed differential form, or sometimes the result ofintegrating such a form..

1. An integral of the first kind is a holomorphic closed differential form.

2. An integral of the second kind is a meromorphic closed differential form with no residues.

3. An integral of the third kind is a meromorphic closed differential form whose poles are all simple.

4. A simple integral is a closed 1-form, or the result of integrating a 1-form.

5. A double integral is a closed 2-form, or the result of integrating a 2-form.

invariant (Noun) A polynomial in the coefficients of a homogeneous form, invariant under some group of lineartransformations. See also covariant, contravariant, concomitant.

inversion An inversion is a transformation of order 2 exchanging the inside and outside of a circle. See Salmon(1879, p.103)

involute An involute is a curve obtained by unrolling a string around a curve. See Salmon (1879, p. 278)

involution 1. A transformation whose square is the identity. Cremona transformations that are involutions includeBertini involutions, Geiser involutions, and De Jonquières involutions.

irregularity The irregularity of a surface is the dimension of the space of holomorphic 1-forms on a non-singularprojective surface; see Hodge number.

isologue Given a Cremoma transformation T, the isologue of a point p is the set of points x such that p, x, T(x) arecollinear. The point p is called the center of the isologue.

3.12 J

Jacobian 1. The Jacobian variety of a curve

2. A Jacobian curve; see below

Jacobian curve The locus of double points of curves of a net. (Semple & Roth 1949, p.115)

Jacobian set The set of free double points of a pencil of curves. (Semple & Roth 1949, p.119)

Jacobian system The linear system generated by Jacobian curves. (Semple & Roth 1949, p.117)

join The join of two linear spaces is the smallest linear space containing both of them.

3.13. K 43

3.13 K

kenotheme An intersection of n hypersurfaces in n-dimensional projective space. (Sylvester 1853, Glossary p. 543–548) Archaic.

keratoid Horn-like. A keratoid cusp is one whose two branches curve in opposite direction; see ramphoid cusp.Salmon (1879)

Kirkman point One of the 60 points lying on 3 of the Plücker lines associated with 6 points on a conic.

Klein surface A Klein icosahedral surface, a cubic surface

Kronecker index The intersection number of two curves on a surface

Kummer surface Main article: Kummer surfaceA quartic surface with 16 nodes

3.14 L

Laguerre net A net V of plane curves of some degree d such that the base locus of a generic pencil of V is the baselocus of V together with d–1 collinear points (Dolgachev 2012, theorem 7.3.5) (Coolidge 1931, p. 423)

lemniscate A lemniscate is a curve resembling a figure 8. See Salmon (1879, p.42)

limaçon A limaçon is a curve traced by a point on a circle rolling around a similar circle. See Salmon (1879, p.43)

line A line in projective space; in other words a subvariety of degree 1 and dimension 1.

line coordinates Projective coordinates. See Salmon (1879, p. 7)

linear Degree 1

linear system A linear system of divisors, given by the zeros of elements of a vector space of sections of a linebundle

locus 1-A subset of projective space given by points satisfying some condition

3.15 M

manifold An algebraic manifold is a cycle of projective space, in other words a formal linear combination of irre-ducible subvarieties. Algebraic manifolds may have singularities, so their underlying topological spaces neednot be manifolds in the sense of differential topology. Semple & Roth (1949, p.14–15)

meet The meet of two sets is their intersection.

Möbius tetrads Main article: Möbius configurationTwo tetrads such that the plane containing any three points of one tetrad contains a point of the other. (Baker1922, p.62)

model 1. A variety whose points (or sometimes hyperplane sections) correspond to elements of some family. Similarto what is now called a parameter space or moduli space.

2. A model for a field extension K of a field k is a projective variety over k together with an isomorphism betweenK and its field of rational functions.

modulus A function of algebraic varieties depending only on the isomorphism type; in other words, a function on amoduli space

Moebius tetrads See #Möbius tetrads

monoid A surface of degree n with a point of multiplicity n–1. (Semple & Roth 1949, p.187)

44 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

monoidal transformation ACremona transformation of projective space generated by a family of monoids with thesame point of multiplicity n–1. More generally a blow-up along a subvariety, called the center of the monoidaltransformation. (Semple & Roth 1949, p.187)

multiple A multiple point is a singular point (one with a non-regular local ring).

multiplicity The multiplicity of a point on a hypersurface is the degree of the first non-vanishing coefficient ofthe Taylor series at the point. More generally one can define the multiplicity of any point of a variety as themultiplicity of its local ring. A point has multiplicity 1 if and only if it is non-singular.

3.16 NNéron–Severi group The Néron–Severi group is the group of divisors module numerical equivalence.

nest Two components (circuits) of a real algebraic curve are said to nest if one is inside the other. (Coolidge 1931)

net 1. A 2-dimensional linear system. See “pencil” and “web”. See also Laguerre net.

2. A harmonic net is a set of points on a line containing the harmonic conjugate of any point with respect to anyother two points. (Baker 1922, p.133)

Newton polygon Main article: Newton polygonThe convex hull of the points with coordinates given by the exponents of the terms of a polynomial.

nodal A nodal tangent to a singular point of a curve is one of the lines of its tangent cone. (Semple & Roth 1949,p.26)

node A singular point p of a hypersurface f = 0, usually with the determinant of the Hessian of f not zero at p.(Cayley 1852)

node cusp A singularity of a curve where a node and a cusp coincide at the same point. (Salmon 1879, p. 207)

normal 1. A subvariety of projective space is linearly normal if the linear system defining the embedding is complete;see rational normal curve

2. Orthogonal to the tangent space, such as a line orthogonal to the tangent space or the normal bundle.

3. A normal intersection is an intersection with the “expected” codimension (given a sum of codimensions). (SempleRoth, p.16)

4. Local rings are integrally closed; see normal scheme.

null-polarity A correlation given by a skew symmetric matrix. A null-polarity of the projective space of a vectorspace is essentially a non-degenerate skew-symmetric bilinear form, up to multiplication by scalars. See alsopolarity. (Semple & Roth 1949, p.9)

3.17 Ooctad A set of 8 points

octic 1. (Adjective) Degree 8

2. (Noun) A degree 8 projective variety

order 1. Now called degree of an algebraic variety: the number of intersection points with a generic linear subspaceof complementary dimension. (Semple & Roth 1949, p.15)

2. The order of a covariant or concomitant: its degree in the contravariant variables.

3. The order of a Cremona transformation is the order (degree) of its homaloids. (Semple & Roth 1949, p.46)

ordinary An ordinary point of multiplicity m of a curve is one with m distinct tangent lines.

3.18. P 45

oscnode A double point of a plane curve that is also a point of osculation; in other words the two branches meet toorder at least 3. (Cayley 1852)

osculate Kiss; to meet with high order. See Salmon (1879, p. 356)

osculating plane A tangent plane of a space curve having third order contact with it.

outpolar quadric See (Baker 1922, vol 2, p. 33, volume III, p. 52)

3.18 PPappus 1. Pappus of Alexandria.

2. The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus’s hexagon theorem.

parabolic point A point of a variety that also lies in the Hessian.

parallel 1. Meeting at the line or plane at infinity, as in parallel lines

2. A parallel curve is the envelope of a circle of fixed radius moving along another curve. (Coolidge 1931, p.192)

partitivity The number of connected components of a real algebraic curve. See Salmon (1879, p.165)

Pascal Short for Pascal line, the line determined by 6 points of a conic in Pascal’s theorem

pedal The pedal curve of C with respect to a pedal point P is the locus of points X such that the line through Xorthogonal to PX is tangent to C. (Salmon 1879, p.96)

pencil A 1-dimensional linear system. See Lefschetz pencil.

pentad A set of 5 points

pentahedron A union of 5 planes, in particular the Sylvester pentahedron of a cubic surface.

period The integral of a differential form over a submanifold

perspectivity An isomorphism between two projective lines (or ranges) of projective space such that the lines joiningeach point of one line to the corresponding point of the other line all pass through a fixed point, called the centerof the perspectivity or the perspector.

perspector The center of a perspectivity

perspectrix The line in Desargues theorem on which the intersections of pairs of sides of two perspective triangleslie

pinch A pinch point is a singular point of a surface, where the two tangent planes of a point on a double curvecoincide in a double plane, called the pinch plane. (Semple & Roth 1949, p.175)

pippian Introduced by Cayley (1857). Now called the Cayleyan. See also quippian.

Plücker Main article: Julius Plücker

1. For Plücker characteristic see characteristic

2. A Plücker line is one of the 15 lines containing 4 of the 20 Steiner points associated to 6 points on a conic. ThePlücker lines meet in threes at the 60 Kirkman points. (Dolgachev 2012, p.124)

plurigenus Plural plurigenera

The dth plurigenus of a variety is the dimension of the space of sections of the dth power of the canonical linebundle.

point-star A family of lines with a common point

polar 1. (Adjective) Related by a polarity

46 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

2. The polar conic is the zero set of the quadratic form associated to a polarity, or equivalently the set of self-conjugate points of the polarity.

3. (Noun) The first polar, second polar, and so on are varieties of degrees n–1, n–2, ... formed from a point and ahypersurface of degree n by polarizing the equation of the hypersurface.(Semple & Roth 1949, p.11)

4. A polar or polar line is the line corresponding to a point under a polarity of the projective plane.

polarity A correlation given by a symmetrical matrix, or a correlation of period 2. A polarity of the projective spaceof a vector space is essentially a non-degenerate symmetric bilinear form, up to multiplication by scalars. Seealso null-polarity. (Semple & Roth 1949, p.9)

pole 1. The point corresponding to a hyperplane under a polarity.

2. A singularity of a rational function.

poloconic

polocubic

poloquartic The poloconic (also called conic polar) of a line in the plane with respect to a cubic curve is the locusof points whose first polar is tangent to the line. (Dolgachev 2012, p. 156–157)

polygonal A polygonal (or k-gonal) curve is a curve together with a map (of degree k) to the projective line. Thedegree of the map is called the gonality of the curve. When the degree is 1, 2, or 3 the curve is called rational,hyperelliptic, or trigonal.

porism 1. A porism is a corollary, especially in geometry, as in Poncelet’s porism. The precise meaning seems tobe controversial.

2. An arrangement of geometrical figures (such as lines or circles) that are inscribed in one curve and circumscribedaround another, as in Poncelet’s porism or Steiner’s porism. There seems to be some confusion about whether“porism” refers to the geometrical configuration or to the statement of the result.

poristic Having either no solutions or infinitely many (Semple & Roth 1949, p.186). For example, Poncelet’s porismand Steiner’s porism imply that if there is one way to arrange lines or circles then there are infinitely many ways.

postulated A postulated object (point, line, and so on) is an object in some larger space. For example, a point atinfinity of projective space is a postulated point of affine space. (Baker 1922)

postulation The postulation of a variety for some family is the number of independent conditions needed to forcean elements of the family to contain the variety. (Semple & Roth 1949, p.440)

power of a point Laguerre defined the power of a point with respect to an algebraic curve of degree n to be theproduct of the distances from the point to the intersections with a circle through it, divided by the nth powerof the diameter. He showed that this is independent of the choice of circle through the point. (Coolidge 1931,p.176)

prime A hyperplane of projective space. (Semple & Roth 1949, p.1)

primal A projective hypersurface. (Semple & Roth 1949, p.10)

projectivity An isomorphism between two projective lines (or ranges). A projectivity is a product of at most threeperspectivities.

propinquity A number depending on two branches at a point, defined by Coolidge (1931, p. 224).

proximate For proximate points see (Zariski 1935, p.9)

pure All components are of the same dimension. Now called equidimensional. (Semple & Roth 1949, p.15)

3.19. Q 47

3.19 Q

quadratic transformation 1. A Cremona transformation of degree 2. A standard quadratic transformation is onesimilar to the map taking each coordinate to its inverse

2. A monomial transformation with center a point, or in other words a blowup at a point.

quadric Degree 2, especially a degree 2 projective variety. Not to be confused with quantic or quartic.

quadrisecant A quadrisecant is a line meeting something in four points

quadro-cubic, quadro-quartic A quadro-cubic or quadro-quartic transformation is a Cremona transformation suchthat the homaloids of the transformation have degree 2 and those of its inverse have degree 3 or 4. (Semple &Roth 1949, p.180, 188)

quantic A homogeneous polynomial in several variables, now usually called a form. Not to be confused with quarticor quadric.

quarto-quartic A Quarto-quartic transformation is a Cremona transformation such that the homaloids of the trans-formation and its inverse all have degree 4. (Semple & Roth 1949, p.187)

quaternary Depending on four variables, as in quaternary form.

quartic Degree 4, especially a degree 4 projective variety. Not to be confused with quantic or quadric.

quintic Degree 5, especially a degree 5 projective variety

quippian A quippian is a degree 5 class 3 contravariant of a plane cubic introduced by Cayley (1857) and discussedby Dolgachev (2012, p.157). See also pippian.

quotient ring The quotient ring of a point (or more generally a subvariety) is what is now called its local ring, formedby adding inverses to all functions that do not vanish identically on it.

3.20 R

ramphoid Beak-like. A ramphoid cusp is one whose two branches curve in the same direction; see keratoid cusp.Salmon (1879, p.46)

rank 1. The rank of a projective curve is the number of tangents to the curve meeting a generic linear subspace ofcodimension 2. (Semple & Roth 1949, p.84)

2. The rank of a projective surface is the rank of a curve given by the intersection of the surface with a generichyperplane. (Semple & Roth 1949, p.193) See order, class, type.

range 1. The set of all points on a line. (Coxeter 1969, p.242)

2. A labeled or finite ordered set of points on a line.

rational 1. Birational to projective space

2. Defined over the rational numbers.

ray A line, especially one in a family of lines

regular 1. A regular surface is one whose irregularity is zero.

2. Having no singularities; see regular local ring

3. Symmetrical, as in regular polygon, regular polyhedron

4. Defined everywhere, as in regular (birational) map

regulus One of the two pencils of lines on a product of two projective planes or a quadric surface.

48 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

related Two ranges (labeled sets) of points on a line are called related if there is a projectivity taking one range tothe other.

representative manifold A parameter space or moduli space for some family of varieties

residual The residual intersection of two varieties consists of the “non-obvious” part of their intersection.

resultant 1. The resultant of two polynomials, given by the determinant of the Sylvester matrix of two binary forms,that vanishes if they have a common root.

2. A Cremona transformation formed from n correlations of n-dimensional projective space.(Semple & Roth 1949,p.180)

reverse Inverse (of a function or birational map)

ruled Covered by lines, as in ruled surface. See also scroll.

3.21 SSn Projective space of dimension n.

Salmon conic The Salmon conic of a pair of plane conics is the locus of points such that the pairs of tangents to thetwo conics are harmonically conjugate. (Dolgachev 2012, p. 119)

satellite 1. If a line meets a cubic curve in 3 points, the residual intersections of the tangents of these points withthe cubic all lie on a line, called the satellite line of the original line. See Salmon (1879, p. 127)

2. A certain plane curve of degree (n–1)(n–2) constructed from a plane curve of degree n and a generic point.(Coolidge, p. 159–161)

3. For satellite points see (Zariski 1935, p.8). Possibly something to do with base points.

scroll A ruled surface with an embedding into projective space so that the lines of the ruled surface are also lines ofprojective space.

secant 1. A line intersecting a variety in 2 points, or more generally an n-dimensional projective space meeting avariety in n+1 points.

2. A secant variety is the union of the secants of a variety.

secundum An intersection of two primes (hyperplanes) in projective space. (Semple & Roth 1949, p.2)

Segre 1. Named after either Beniamino Segre or Corrado Segre

2. A Segre variety or Segre embedding is the product of two projective spaces, or an embedding of this into a largerprojective space.

3. The Segre cubic is a cubic hypersurface in 4-dimensional projective space.

self-conjugate

self-polar 1. Incident with its image under a polarity. In particular the self-conjugate points of a polarity form thepolar conic.

2. A self-conjugate (or self-polar) triangle (or triad) is a triangle such that each vertex corresponds to the oppositeedge under a polarity.

3. A self-conjugate tetrad is a set of 4 points such that the pole of each side lies on the opposite side. (Dolgachev2012, p.123)

septic

septimic 1. (Adjective) Degree 7

2. (Noun) A degree 7 projective variety

3.21. S 49

3. (Noun) A degree 7 form

sextactic point One of the 27 points of an elliptic curve of order dividing 6 but not 3. (Salmon 1879, p.132)

sextic Degree 6, especially a degree 6 projective variety

simple A simple point of a variety is a non-singular point. More generally a simple subvariety W of a variety V isone with a regular local ring, which means roughly that most points ofW are simple points of V.

singular Special in some way, including but not limited to the current sense of having a singularity

skew Intersecting in a set that is either empty or of the “expected” dimension. For example skew lines in projective3-space do not intersect, while skew planes in projective 4-space intersect in a point.

solid A 3-dimensional linear subspace of projective space, or in other words the 3-dimensional analogue of a point,line, or plane. (Semple & Roth 1949, p.4)

special divisor An effective divisor whose first cohomology group (of the associated invertible sheaf) is non-zero

spinode A cusp. (Cayley 1852), Salmon (1879, p.23)

star A collection of lines (and sometimes planes and so on) with a common point, called the center of the star.(Baker 1922, p.109)

stationary point A cusp. See Salmon (1879, p.23)

Steiner

Steinerian 1. Named after Jakob Steiner

2. A Steinerian is the locus of the singular points of the polar quadrics of a hypersurface. Salmon (1879)

3. A Steiner surface is a certain embedding of the projective plane into projective 3-space.

4. a Steiner point is one of the 20 points lying on 3 of the Pascal lines associated with 6 points on a conic.

Steiner–Hessian One of Cayley’s names for the Cayleyan. See Salmon (1879, p. 352)

surface An abstract surface together with an embedding into projective space.

superabundance of a divisor on a surface. The dimension of the first cohomology group of the correspondingsheaf.

symmetroid The zeros of the determinant of a symmetric matrix of linear forms

syntheme A partition of a set of 6 elements into 3 pairs, or an element of the symmetric group on 6 points of cycleshape 222. (Dolgachev 2012)

system A family of algebraic sets in projective space; for example, a line system is a family of lines.

syzygetic Paired. Opposite of azygetic, meaning unpaired. Example: syzygetic triad, syzygetic tetrad, syzygetic set,syzygetic pencil.

syzygy 1. A point is in syzygy with some other points if it is in the linear subspace generated by them.(Baker 1922,vol I, p. 33) A syzygy is a linear relation between points in an affine space.

2. An algebraic relation between generators of a ring, especially a ring of invariants or covariants.

3. A linear relation between generators of a module, or more generally an element of the kernel of a homomorphismof modules.

4. A global syzygy is a resolution of a module or sheaf.

50 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

3.22 Ttacnode A tacnode is a point of a curve where two branches meet in the same direction. (Cayley 1852)

tacnode-cusp A singularity of a plane curve where a tacnode and a cusp are combined at the same point.(Salmon1879, p.207)

tact-invariant An invariant of two curves that vanishes if they touch each other. See Salmon (1879, p.76)

tangent cone A tangent cone is a cone defined by the non-zero terms of smallest degree in the Taylor series at apoint of a hypersurface.

tangential equation The tangential equation of a plane curve is an equation giving the condition for a line to betangent to the curve. In other words it is the equation of the dual curve. It is not the equation of a tangent to acurve.

ternary Depending on three variables, as in ternary form

tetrad A set of 4 points

tetragram Synonym for complete quadrilateral

tetrahedroid A tetrahedroid is a special kind of Kummer surface.

tetrahedron A geometric configuration consisting of 4 points and the 6 lines joining pairs. This is similar to thelines and infinite edges of a polyhedral tetrahedron, but in algebraic geometry one sometimes does not includethe faces of the tetrahedron.

tetrastigm Synonym for complete quadrangle

threefold 1. (Adjective) Three-dimensional

2. (Noun) A 3-dimensional variety

torsal generator. A generator of a scroll (ruled surface) that meets its consecutive generator. See (Semple & Roth1949, p.204)

torse Developable surface.

transvectant An invariant depending on two forms.

transversal A line meeting several other lines. For example, 4 generic lines in projective 3-space have 2 transversalsmeeting all of them.

triad A set of 3 points

tricircular A tricircular curve is one that passes through the circular points at infinity with order 3.

tricuspidal Having three cusps

trigonal A trigonal curve is one with a degree three map to the projective line. See hyperelliptic.

trihedral A set of 3 planes A Steiner trihedral is a set of three tritangent planes of a cubic surface whose intersectionpoint is not on the surface. (Semple & Roth 1949, p.152)

trilinear coordinates Coordinates based on distance from sides of a triangle:Trilinear coordinates.

trinodal Having three nodes

tripartite Having three connected components. Salmon (1879, p.165)

trisecant A line meeting a variety in 3 points. See trisecant identity.

tritangent Meeting something in 3 tangent points, such as a tritangent conic to a cubic curve or a tritangent planeof a cubic surface.

trope A trope is a singular (meaning special) tangent space.(Cayley 1869, p.202) The word is mostly used for atangent space of a Kummer surface touching it along a conic.

3.23. U 51

twisted A twisted cubic is a degree 3 embedding of the projective line in projective 3-space

total A set of 5 partitions of a 6-element set into three pairs, such that no two elements of the total have a pair incommon. For example, {(12)(36)(45), (13)(24)(56), (14)(26)(35), (15)(23)(46), (16)(25)(34)} (Dolgachev2012)

type The type of a projective surface is the number of tangent planes meeting a generic linear subspace of codimen-sion 4. (Semple & Roth 1949, p.193)

3.23 Uundulation A point of undulation of a curve is where the tangent meets the curve to fourth order; also called a

hyperflex. See inflection point. (Salmon 1879, p.35, 211)

unibranch Having only one branch at a point. For example, a cusp of a plane curve is unibranch, while a node isnot.

unicursal A unicursal curve is one that is rational, in other words birational to the projective line. See Salmon (1879,p. 29)

unipartite Connected. See Salmon (1879, p.165)

unirational 1. A correspondence is called unirational if it is generically injective, in other words a rational map.(Semple & Roth 1949, p.20)

2. A variety is called unirational if it is finitely covered by a rational variety.

united point A point in the intersection of the diagonal and a correspondence from a set to itself.

unode A double point of a surface whose tangent cone consists of one double plane. See binode.

3.24 Vvalence

valency The valence or valency of a correspondence T on a curve is a number k such that the divisors T(P)+kP areall linearly equivalent. A correspondence need not have a valency. Semple & Roth (1949, p.368)

Veronese surface Main article: Veronese surfaceAn embedding of the projective plane in 5-dimensional projective space

virtual An estimate for something that is often but not always correct, such as virtual genus, virtual dimension, andso on. If some number is given by the dimension of a space of sections of some sheaf, the correspondingvirtual number is sometimes given by the corresponding Euler characteristic, and equal to the dimension whenall higher cohomology groups vanish. See superabundance.

3.25 Wweb A 3-dimensional linear system. See “net” and “pencil”. (Semple & Roth 1949, p.160)

Weddle surface Main article: Weddle surfaceA quartic surface in projective space given by the locus of the vertex of a cone passing through 6 points ingeneral position.

Weierstrass point Main article: Weierstrass pointA point on a curve where the dimension of the space of rational functions whose only singularity is a pole ofsome order at the point is higher than normal.

Wirtinger sextic Main article: Wirtinger sexticA degree 4 genus 6 plane curve with nodes at the 6 points of a complete quadrangle.

52 CHAPTER 3. GLOSSARY OF CLASSICAL ALGEBRAIC GEOMETRY

3.26 XYZZeuthen–Segre invariant The Zeuthen–Segre invariant is essentially the Euler characteristic of a non-singular pro-

jective surface.

3.27 See also• Glossary of algebraic geometry

• Glossary of arithmetic and Diophantine geometry

• Glossary of commutative algebra

• Glossary of differential geometry and topology

• Glossary of invariant theory

• Glossary of Riemannian and metric geometry

• Glossary of scheme theory

• List of complex and algebraic surfaces

• List of surfaces

• List of curves

3.28 References• Baker, Henry Frederick (1922), Principles of geometry. Volume 1. Foundations, Cambridge Library Collection,Cambridge University Press, ISBN 978-1-108-01777-0, MR 2849917 Reprinted 2010

• Baker, Henry Frederick (1922), Principles of geometry. Volume 2. Plane geometry, Conics, circles, non-Euclidean geometry, Cambridge Library Collection, Cambridge University Press, ISBN 978-1-108-01778-7,MR 2857757 Reprinted 2010

• Baker, Henry Frederick (1923), Principles of geometry. Volume 3. Solid geometry. Quadrics, cubic curves inspace, cubic surfaces., Cambridge Library Collection, Cambridge University Press, ISBN 978-1-108-01779-4,MR 2857520 Reprinted 2010

• Baker, Henry Frederick (1925), Principles of geometry. Volume 4. Higher geometry. Being illustrations of theutility of the consideration of higher space, especially of four and five dimensions, Cambridge Library Collection,Cambridge University Press, ISBN 978-1-108-01780-0, MR 2849669 Reprinted 2010

• Baker, Henry Frederick (1933), Principles of geometry. Volume 5. Analytical principals of the theory ofcurves, Cambridge Library Collection, Cambridge University Press, ISBN 978-1-108-01781-7, MR 2850139Reprinted 2010

• Baker, Henry Frederick (1933), Principles of geometry. Volume 6. Introduction to the theory of algebraicsurfaces and higher loci., Cambridge Library Collection, Cambridge University Press, ISBN 978-1-108-01782-4, MR 2850141 Reprinted 2010

• Cayley, Arthur (1852), “On the singularities of surfaces”, The Cambridge and Dublin Mathematical Journal 7:166

• Cayley, Arthur (1857), “A Memoir on Curves of the Third Order”, Philosophical Transactions of the RoyalSociety of London (The Royal Society) 147: 415–446, doi:10.1098/rstl.1857.0021, ISSN 0080-4614, JSTOR108626

• Cayley, Arthur (1869), “A Memoir on the Theory of Reciprocal Surfaces”, Philosophical Transactions of theRoyal Society of London (The Royal Society) 159: 201–229, doi:10.1098/rstl.1869.0009, ISSN 0080-4614,JSTOR 108996

3.28. REFERENCES 53

• Coolidge, Julian Lowell (1931), A treatise on algebraic plane curves, Oxford University Press, ISBN 978-0-486-49576-7, MR 0120551

• Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: JohnWiley & Sons,ISBN 978-0-471-50458-0, MR 123930

• Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view (PDF), Cambridge University Press,ISBN 978-1-107-01765-8

• Hudson, R. W. H. T. (1990), Kummer’s quartic surface, Cambridge Mathematical Library, Cambridge Uni-versity Press, ISBN 978-0-521-39790-2, MR 1097176

• Jessop, Charles Minshall (1916), Quartic surfaces with singular points, Cambridge University Press, ISBN978-1-112-28262-1

• Salmon, George (1879) [1852], A treatise on the higher plane curves, New York: Hodges, Foster and Figgis,ISBN 978-1-4181-8252-6, MR 0115124

• Schubert, Hermann (1886), “Die n-dimensionalen Verallgemeinerungen der fundamentalen Anzahlen unseresRaums”, Mathematische Annalen (Springer Berlin / Heidelberg) 26: 26–51, doi:10.1007/BF01443568, ISSN0025-5831

• Semple, J. G.; Roth, L. (1949), Introduction to algebraic geometry, Oxford Science Publications, The ClarendonPress Oxford University Press, ISBN 978-0-19-853363-4, MR 814690

• Sylvester, James Joseph (1853), “On a Theory of the Syzygetic Relations of Two Rational Integral Functions,Comprising an Application to the Theory of Sturm’s Functions, and That of the Greatest Algebraical CommonMeasure”, Philosophical Transactions of the Royal Society of London (The Royal Society) 143: 407–548,doi:10.1098/rstl.1853.0018, ISSN 0080-4614, JSTOR 108572

• Zariski, Oscar (1935), Algebraic surfaces, Classics in Mathematics, Berlin, New York: Springer-Verlag, ISBN978-3-540-58658-6, MR 1336146

Chapter 4

Glossary of cryptographic keys

This glossary lists types of keys as the term is used in cryptography, as opposed to door locks. Terms that are primarilyused by the U.S. National Security Agency are marked (NSA). For classification of keys according to their usage seecryptographic key types.

• 40-bit key - key with a length of 40 bits, once the upper limit of what could be exported from the U.S. andother countries without a license. Considered very insecure. See key size for a discussion of this and otherlengths.

• authentication key - Key used in a keyed-hash message authentication code, or HMAC.

• benign key - (NSA) a key that has been protected by encryption or other means so that it can be distributedwithout fear of its being stolen. Also called BLACK key.

• content-encryption key (CEK) a key that may be further encrypted using a KEK, where the content may bea message, audio, image, video, executable code, etc.

• cryptovariable - NSA calls the output of a stream cipher a key or key stream. It often uses the term crypto-variable for the bits that control the stream cipher, what the public cryptographic community calls a key.

• derived key - keys computed by applying a predetermined hash algorithm or key derivation function to apassword or, better, a passphrase.

• electronic key - (NSA) key that is distributed in electronic (as opposed to paper) form. See EKMS.

• ephemeral key - A key that only exists within the lifetime of a communication session.

• expired key - Key that was issued for a use in a limited time frame (cryptoperiod in NSA parlance) which haspassed and, hence, the key is no longer valid.

• key encryption key (KEK) - key used to protect other keys (e.g. TEK, TSK).

• key production key (KPK) -Key used to initialize a keystream generator for the production of other electron-ically generated keys.

• key fill - (NSA) loading keys into a cryptographic device. See fill device.

• FIREFLY key - (NSA) keys used in an NSA system based on public key cryptography.

• master key - key from which all other keys (or a large group of keys) can be derived. Analogous to a physicalkey that can open all the doors in a building.

54

55

• message encryption key (MEK) - See traffic encryption key.

• one-time pad - keying material that is as long as the plaintext and only used once. See one-time pad article.

• paper key - (NSA) keys that are distributed in paper form, such as printed lists of settings for rotor machines,or keys in punched card or paper tape formats. Paper key is easily copied. SeeWalker spy ring, RED key.

• poem key - Keys used by OSS agents in World War II in the form of a poem that was easy to remember. SeeLeo Marks.

• Public/private key - in public key cryptography, separate keys are used to encrypt and decrypt a message. Theencryption key (public key) need not be kept secret and can be published. The decryption or private keymustbe kept secret to maintain confidentiality. Public keys are often distributed in a signed public key certificate.

• pre-placed key - (NSA) large numbers of keys (perhaps a year’s supply) that are loaded into an encryptiondevice allowing frequent key change without refill.

• RED key - (NSA) symmetric key in a format that can be easily copied, e.g. paper key or unencrypted electronickey. Opposite of BLACK or benign key.

• revoked key - a public key that should no longer be used, typically because its owner is no longer in the role forwhich it was issued or because it may have been compromised. Such keys are placed on a certificate revocationlist or CRL.

• session key - key used for one message or an entire communications session. See traffic encryption key.

• symmetric key - a key that is used both to encrypt and decrypt a message. Symmetric keys are typically usedwith a cipher and must be kept secret to maintain confidentiality.

• traffic encryption key (TEK) - a symmetric key that is used to encrypt messages. TEKs are typically changedfrequently, in some systems daily and in others for every message. See session key.

• transmission security key (TSK) - (NSA) seed for a pseudorandom number generator that is used to controla radio in frequency hopping or direct-sequence spread spectrum modes. See SINCGARS, electronic warfare.

• seed key - (NSA) a key used to initialize a cryptographic device so it can accept operational keys using benigntransfer techniques. Also a key used to initialize a pseudorandom number generator to generate other keys.

• signature key - public key cryptography can also be used to electronically sign messages. The private key isused to create the electronic signature, the public key is used to verify the signature. Separate public/privatekey pairs must be used for signing and encryption. The former is called signature keys.

• stream key - the output of a stream cipher as opposed to the key (or cryptovariable in NSA parlance) thatcontrols the cipher

• training key - (NSA) unclassified key used for instruction and practice exercises.

• Type 1 key - (NSA) keys used to protect classified information. See Type 1 product.

• Type 2 key - (NSA) keys used to protect sensitive but unclassified (SBU) information. See Type 2 product.

• Vernam key - Type of key invented by Gilbert Vernam in 1918. See stream key.

• zeroized key - key that has been erased (see zeroisation.)

56 CHAPTER 4. GLOSSARY OF CRYPTOGRAPHIC KEYS

4.1 References• Schneier, Bruce. Applied Cryptography, Second Edition, John Wiley & Sons, 1996. ISBN 0-471-11709-9

• National Information Assurance (IA) Glossary, Committee on National Security Systems, CNSS InstructionNo. 4009, 2010.

• Link 16 Joint Key Management Plan, CJCSM 6520.01A, 2011

4.2 See also• Specific encryption systems and ciphers have key types associated with them, e.g. PGP key, DES key, AESkey, RC4 key, BATON key, Kerberos key, etc.

• Category:Cryptographic algorithms

• Category:Cryptographic protocols

Chapter 5

Glossary of differential geometry andtopology

This is a glossary of terms specific to differential geometry and differential topology. The following two glossariesare closely related:

• Glossary of general topology

• Glossary of Riemannian and metric geometry.

See also:

• List of differential geometry topics

Words in italics denote a self-reference to this glossary.Contents :

• Top

• 0–9

• A

• B

• C

• D

• E

• F

• G

• H

• I

• J

• K

• L

• M

57

58 CHAPTER 5. GLOSSARY OF DIFFERENTIAL GEOMETRY AND TOPOLOGY

• N

• O

• P

• Q

• R

• S

• T

• U

• V

• W

• X

• Y

• Z

5.1 A

Atlas

5.2 B

Bundle, see fiber bundle.

5.3 C

ChartCobordismCodimension. The codimension of a submanifold is the dimension of the ambient space minus the dimension of thesubmanifold.Connected sumConnectionCotangent bundle, the vector bundle of cotangent spaces on a manifold.Cotangent space

5.4 D

Diffeomorphism. Given two differentiable manifolds M and N, a bijective map f from M to N is called a diffeo-morphism if both f : M → N and its inverse f−1 : N → M are smooth functions.Doubling, given a manifold M with boundary, doubling is taking two copies of M and identifying their boundaries.As the result we get a manifold without boundary.

5.5. E 59

5.5 E

Embedding

5.6 F

Fiber. In a fiber bundle, π: E → B the preimage π−1(x) of a point x in the base B is called the fiber over x, oftendenoted Ex.Fiber bundleFrame. A frame at a point of a differentiable manifold M is a basis of the tangent space at the point.Frame bundle, the principal bundle of frames on a smooth manifold.Flow

5.7 G

Genus

5.8 H

Hypersurface. A hypersurface is a submanifold of codimension one.

5.9 I

Immersion

5.10 L

Lens space. A lens space is a quotient of the 3-sphere (or (2n + 1)-sphere) by a free isometric action of Z .

5.11 M

Manifold. A topological manifold is a locally Euclidean Hausdorff space. (In Wikipedia, a manifold need not beparacompact or second-countable.) A Ck manifold is a differentiable manifold whose chart overlap functions are ktimes continuously differentiable. A C∞ or smooth manifold is a differentiable manifold whose chart overlap functionsare infinitely continuously differentiable.

5.12 N

Neat submanifold. A submanifold whose boundary equals its intersection with the boundary of the manifold intowhich it is embedded.

5.13 P

Parallelizable. A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to thetangent bundle being trivial.

60 CHAPTER 5. GLOSSARY OF DIFFERENTIAL GEOMETRY AND TOPOLOGY

Principal bundle. A principal bundle is a fiber bundle P → B together with an action on P by a Lie group G thatpreserves the fibers of P and acts simply transitively on those fibers.Pullback

5.14 S

SectionSubmanifold, the image of a smooth embedding of a manifold.SubmersionSurface, a two-dimensional manifold or submanifold.Systole, least length of a noncontractible loop.

5.15 T

Tangent bundle, the vector bundle of tangent spaces on a differentiable manifold.Tangent field, a section of the tangent bundle. Also called a vector field.Tangent spaceTorusTransversality. Two submanifolds M and N intersect transversally if at each point of intersection p their tangentspaces Tp(M) and Tp(N) generate the whole tangent space at p of the total manifold.Trivialization

5.16 V

Vector bundle, a fiber bundle whose fibers are vector spaces and whose transition functions are linear maps.Vector field, a section of a vector bundle. More specifically, a vector field can mean a section of the tangent bundle.

5.17 W

Whitney sum. A Whitney sum is an analog of the direct product for vector bundles. Given two vector bundles α andβ over the same base B their cartesian product is a vector bundle over B ×B. The diagonal map B → B ×B inducesa vector bundle over B called the Whitney sum of these vector bundles and denoted by α⊕β.

Chapter 6

Glossary of experimental design

The following is a glossary of terms. It is not intended to be all-inclusive.

6.1 Concerned fields

• Statistics

• Experimental design

• Estimation theory

6.2 Glossary

• Alias: When the estimate of an effect also includes the influence of one or more other effects (usually highorder interactions) the effects are said to be aliased (see confounding). For example, if the estimate of effectD in a four factor experiment actually estimates (D + ABC), then the main effect D is aliased with the 3-wayinteraction ABC. Note: This causes no difficulty when the higher order interaction is either non-existent orinsignificant.

• Analysis of variance (ANOVA): A mathematical process for separating the variability of a group of observa-tions into assignable causes and setting up various significance tests.

• Balanced design: An experimental design where all cells (i.e. treatment combinations) have the same numberof observations.

• Blocking: A schedule for conducting treatment combinations in an experimental study such that any effects onthe experimental results due to a known change in raw materials, operators, machines, etc., become concen-trated in the levels of the blocking variable. Note: the reason for blocking is to isolate a systematic effect andprevent it from obscuring the main effects. Blocking is achieved by restricting randomization.

• Center Points: Points at the center value of all factor ranges.

• Coding Factor Levels: Transforming the scale of measurement for a factor so that the high value becomes +1and the low value becomes −1 (see scaling). After coding all factors in a 2-level full factorial experiment, thedesign matrix has all orthogonal columns. Coding is a simple linear transformation of the original measurementscale. If the “high” value is X and the “low” value is XL (in the original scale), then the scaling transformationtakes any original X value and converts it to (X − a)/b, where a = (X + XL)/2 and b = (X −XL)/2. To goback to the original measurement scale, just take the coded value and multiply it by b and add a or, X = b ×(coded value) + a. As an example, if the factor is temperature and the high setting is 65°C and the low settingis 55°C, then a = (65 + 55)/2 = 60 and b = (65 − 55)/2 = 5. The center point (where the coded value is 0) hasa temperature of 5(0) + 60 = 60°C.

61

62 CHAPTER 6. GLOSSARY OF EXPERIMENTAL DESIGN

• Comparative design: A design that allows the (typically mean-unbiased) estimation of the difference in factoreffects, especially for the difference in treatment effects. The estimation of differences between treatmenteffects can be made with greater reliability than the estimation of absolute treatment effects.

• Confounding: A confounding design is one where some treatment effects (main or interactions) are estimatedby the same linear combination of the experimental observations as some blocking effects. In this case, thetreatment effect and the blocking effect are said to be confounded. Confounding is also used as a general termto indicate that the value of a main effect estimate comes from both the main effect itself and also contaminationor bias from higher order interactions. Note: Confounding designs naturally arise when full factorial designshave to be run in blocks and the block size is smaller than the number of different treatment combinations.They also occur whenever a fractional factorial design is chosen instead of a full factorial design.

• Control group: a set of experimental units to which incidental treatments are applied but not main treatments.For example, in applying a herbicide as one treatment, plots receiving that treatment might be driven overby a machine applying the herbicide but treatments not receiving the herbicide would not normally be drivenover. The machine traffic is an incidental treatment. If there was a concern that the machine traffic mighthave an effect on the variable being measured (e.g. death of strawberry plants), then a control treatmentwould receive the machine traffic but no herbicide. Control groups are a way of eliminating the possibilityof incidental treatments being the cause of measured effects. The incidental treatments are controlled for.Compare treatment groups. A treatment that is only the absence of the manipulation being studied is simplyone of the treatments and not a control, though it is now common to refer to a non-manipulated treatment as acontrol.

• Crossed factors: See factors below.

• Design: A set of experimental runs which allows you to fit a particular model and estimate your desired effects.

• Designmatrix: Amatrix description of an experiment that is useful for constructing and analyzing experiments.

• Effect (of a factor): How changing the settings of a factor changes the response. The effect of a single factor isalso called a main effect. A treatment effect may be assumed to be the same for each experimental unit, by theassumption of treatment-unit additivity; more generally, the treatment effect may be the average effect. Othereffects may be block effects. (For a factor A with two levels, scaled so that low = −1 and high = +1, the effectof A has a mean-unbiased estimator that is evaluated by subtracting the average observed response when Ais −1 from the average observed response when A = +1 and dividing the result by 2; division by 2 is neededbecause the −1 level is 2 scaled units away from the +1 level.)

• Error: Unexplained variation in a collection of observations. See Errors and residuals in statistics. Note:experimental designs typically require understanding of both random error and lack of fit error.

• Experimental unit: The entity to which a specific treatment combination is applied. For example, an experi-mental unit can be a

• PC board• silicon wafer• tray of components simultaneously treated• individual agricultural plants• plot of land• automotive transmissions• etc.

• Factors: Process inputs that an investigator manipulates to cause a corresponding change in the output. Somefactors cannot be controlled by the experimenter but may affect the responses. These uncontrolled factorsshould be measured and used in the data analysis, if their effect is significant. Note: The inputs can be discreteor continuous.

• Crossed factors: Two factors are crossed if every level of one occurs with every level of the other in theexperiment.

• Nested factors: A factor “A” is nested within another factor “B” if the levels or values of “A” are differentfor every level or value of “B”. Note: Nested factors or effects have a hierarchical relationship.

6.2. GLOSSARY 63

• Fixed effect: An effect associated with an input variable that has a limited number of levels or in which only alimited number of levels are of interest to the experimenter.

• Interaction: Occurs when the effect of one factor on a response depends on the level of another factor(s).

• Lack of fit error: Error that occurs when the analysis omits one or more important terms or factors from theprocess model. Note: Including replication in a designed experiment allows separation of experimental errorinto its components: lack of fit and random (pure) error.

• Model: Mathematical relationship which relates changes in a given response to changes in one or more factors.

• Nested Factors: See factors above.

• Orthogonality: Two vectors of the same length are orthogonal if the sum of the products of their correspondingelements is 0. Note: An experimental design is orthogonal if the effects of any factor balance out (sum to zero)across the effects of the other factors.

• Random effect: An effect associated with input variables chosen at random from a population having a largeor infinite number of possible values.

• Random error: Error that occurs due to natural variation in the process. Note: Random error is typicallyassumed to be normally distributed with zero mean and a constant variance. Note: Random error is also calledexperimental error.

• Randomization: A schedule for allocating treatment material and for conducting treatment combinations in adesigned experiment such that the conditions in one run neither depend on the conditions of the previous runnor predict the conditions in the subsequent runs. Note: The importance of randomization cannot be overstressed. Randomization is necessary for conclusions drawn from the experiment to be correct, unambiguousand defensible.

• Regression discontinuity design: A design in which assignment to a treatment is determined at least partly bythe value of an observed covariate lying on either side of a fixed threshold.

• Replication: Performing the same treatment combination more than once. Note: Including replication allowsan estimate of the random error independent of any lack of fit error.

• Resolution: In fractional factorial designs, “resolution” describes the degree to which the estimatedmain-effectsare aliased (or confounded) with estimated higher-order interactions (2-level interactions, 3-level interactions,etc.). In general, the resolution of a design is one more than the smallest order interaction which is aliasedwith some main effect. If some main effects are confounded with some 2-level interactions, the resolution is 3.Note: Full factorial designs have no confounding and are said to have resolution “infinity”. For most practicalpurposes, a resolution 5 design is excellent and a resolution 4 design may be adequate. Resolution 3 designsare useful as economical screening designs.

• Response(s): The output(s) of a process. Sometimes called dependent variable(s).

• Response surface: A designed experiment that models the quantitative response, especially for the short-termgoal of improving a process and the longer-term goal of finding optimum factor-values. Traditionally, response-surfaces have been modeled with quadratic-polynomials, whose estimation requires that every factor have threelevels.

• Rotatability: A design is rotatable if the variance of the predicted response at any point x depends only on thedistance of x from the design center point. A design with this property can be rotated around its center pointwithout changing the prediction variance at x. Note: Rotatability is a desirable property for response surfacedesigns (i.e. quadratic model designs).

• Scaling factor levels: Transforming factor levels so that the high value becomes +1 and the low value becomes−1.

• Screening design: A designed experiment that identifies which of many factors have a significant effect on theresponse. Note: Typically screening designs have more than 5 factors.

• Test plan: a written document that gives a specific listing of the test procedures and sequence to be followed.

64 CHAPTER 6. GLOSSARY OF EXPERIMENTAL DESIGN

• Treatment: A treatment is a specific combination of factor levels whose effect is to be compared with othertreatments.

• Treatment combination: The combination of the settings of several factors in a given experimental trial. Alsoknown as a run.

• Treatment group: see Control group

• Variance components: Partitioning of the overall variation into assignable components.

6.3 See also• Glossary of probability and statistics

• Notation in probability and statistics

• List of statistical topics

6.4 External links• “A Glossary of DOE Terminology”, NIST/SEMATECH e-Handbook of Statistical Methods, retrieved 20 March2013

This article incorporates public domain material from websites or documents of the National Institute of Standardsand Technology.

Chapter 7

Glossary of field theory

Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject.(See field theory (physics) for the unrelated field theories in physics.)

7.1 Definition of a field

A field is a commutative ring (F,+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a fieldwe thus can perform the operations addition, subtraction, multiplication, and division.The non-zero elements of a field F form an abelian group under multiplication; this group is typically denoted by F×;The ring of polynomials in the variable x with coefficients in F is denoted by F[x].

7.2 Basic definitionsCharacteristic The characteristic of the field F is the smallest positive integer n such that n·1 = 0; here n·1 stands

for n summands 1 + 1 + 1 + ... + 1. If no such n exists, we say the characteristic is zero. Every non-zerocharacteristic is a prime number. For example, the rational numbers, the real numbers and the p-adic numbershave characteristic 0, while the finite field Zp has characteristic p.

Subfield A subfield of a field F is a subset of F which is closed under the field operation + and * of F and which,with these operations, forms itself a field.

Prime field The prime field of the field F is the unique smallest subfield of F.

Extension field If F is a subfield of E then E is an extension field of F. We then also say that E/F is a field extension.

Degree of an extension Given an extension E/F, the field E can be considered as a vector space over the field F,and the dimension of this vector space is the degree of the extension, denoted by [E : F].

Finite extension A finite extension is a field extension whose degree is finite.

Algebraic extension If an element α of an extension field E over F is the root of a non-zero polynomial in F[x],then α is algebraic over F. If every element of E is algebraic over F, then E/F is an algebraic extension.

Generating set Given a field extension E/F and a subset S of E, we write F(S) for the smallest subfield of E thatcontains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations+,−,*,/ on the elements of F and S. If E = F(S) we say that E is generated by S over F.

Primitive element An element α of an extension field E over a field F is called a primitive element if E=F(α), thesmallest extension field containing α. Such an extension is called a simple extension.

65

66 CHAPTER 7. GLOSSARY OF FIELD THEORY

Splitting field A field extension generated by the complete factorisation of a polynomial.

Normal extension A field extension generated by the complete factorisation of a set of polynomials.

Separable extension An extension generated by roots of separable polynomials.

Perfect field A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields,are perfect.

Imperfect degree Let F be a field of characteristic p>0; then Fp is a subfield. The degree [F:Fp] is called theimperfect degree of F. The field F is perfect if and only if its imperfect degree is 1. For example, if F is afunction field of n variables over a finite field of characteristic p>0, then its imperfect degree is pn.[1]

Algebraically closed field A field F is algebraically closed if every polynomial in F[x] has a root in F; equivalently:every polynomial in F[x] is a product of linear factors.

Algebraic closure An algebraic closure of a field F is an algebraic extension of F which is algebraically closed.Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F.

Transcendental Those elements of an extension field of F that are not algebraic over F are transcendental over F.

Algebraically independent elements Elements of an extension field of F are algebraically independent over F ifthey don't satisfy any non-zero polynomial equation with coefficients in F.

Transcendence degree The number of algebraically independent transcendental elements in a field extension. It isused to define the dimension of an algebraic variety.

7.3 Homomorphisms

Field homomorphism A field homomorphism between two fields E and F is a function

f : E → F

such that

f(x + y) = f(x) + f(y)

and

f(xy) = f(x) f(y)

for all x, y in E, as well as f(1) = 1. These properties imply that f(0) = 0, f(x−1) = f(x)−1 for x in E with x ≠ 0, andthat f is injective. Fields, together with these homomorphisms, form a category. Two fields E and F are calledisomorphic if there exists a bijective homomorphism

f : E → F.

The two fields are then identical for all practical purposes; however, not necessarily in a unique way. See, forexample, complex conjugation.

7.4 Types of fields

Finite field A field with finitely many elements.

Ordered field A field with a total order compatible with its operations.

7.5. FIELD EXTENSIONS 67

Rational numbers

Real numbers

Complex numbers

Number field Finite extension of the field of rational numbers.

Algebraic numbers The field of algebraic numbers is the smallest algebraically closed extension of the field ofrational numbers. Their detailed properties are studied in algebraic number theory.

Quadratic field A degree-two extension of the rational numbers.

Cyclotomic field An extension of the rational numbers generated by a root of unity.

Totally real field A number field generated by a root of a polynomial, having all its roots real numbers.

Formally real field

Real closed field

Global field A number field or a function field of one variable over a finite field.

Local field A completion of some global field (w.r.t. a prime of the integer ring).

Complete field A field complete w.r.t. to some valuation.

Pseudo algebraically closed field A field in which every variety has a rational point.[2]

Henselian field A field satisfying Hensel lemma w.r.t. some valuation. A generalization of complete fields.

Hilbertian field A field satisfying Hilbert’s irreducibility theorem: formally, one for which the projective line is notthin in the sense of Serre.[3][4]

Kroneckerian field A totally real algebraic number field or a totally imaginary quadratic extension of a totally realfield.[5]

CM-field or J-field An algebraic number field which is a totally imaginary quadratic extension of a totally realfield.[6]

Linked field A field over which no biquaternion algebra is a division algebra.[7]

Frobenius field A pseudo algebraically closed field whose absolute Galois group has the embedding property.[8]

7.5 Field extensions

Let E / F be a field extension.

Algebraic extension An extension in which every element of E is algebraic over F.

Simple extension An extension which is generated by a single element, called a primitive element, or generatingelement.[9] The primitive element theorem classifies such extensions.[10]

68 CHAPTER 7. GLOSSARY OF FIELD THEORY

Normal extension An extension that splits a family of polynomials: every root of the minimal polynomial of anelement of E over F is also in E.

Separable extension An algebraic extension in which the minimal polynomial of every element of E over F is aseparable polynomial, that is, has distinct roots.[11]

Galois extension A normal, separable field extension.

Primary extension An extension E/F such that the algebraic closure of F in E is purely inseparable over F; equiv-alently, E is linearly disjoint from the separable closure of F.[12]

Purely transcendental extension An extension E/F in which every element of E not in F is transcendental overF.[13][14]

Regular extension An extension E/F such that E is separable over F and F is algebraically closed in E.[12]

Simple radical extension A simple extension E/F generated by a single element α satisfying αn = b for an elementb of F. In characteristic p, we also take an extension by a root of an Artin–Schreier polynomial to be a simpleradical extension.[15]

Radical extension A tower F = F0 < F1 < · · · < Fk = E where each extension Fi/Fi−1 is a simple radicalextension.[15]

Self-regular extension An extension E/F such that E⊗FE is an integral domain.[16]

Totally transcendental extension An extension E/F such that F is algebraically closed in F.[14]

Distinguished class A class C of field extensions with the three properties[17]

1. If E is a C-extension of F and F is a C-extension of K then E is a C-extension of K.2. If E and F are C-extensions of K in a common overfieldM, then the compositum EF is a C-extension of

K.3. If E is a C-extension of F and E>K>F then E is a C-extension of K.

7.6 Galois theory

Galois extension A normal, separable field extension.

Galois group The automorphism group of a Galois extension. When it is a finite extension, this is a finite group oforder equal to the degree of the extension. Galois groups for infinite extensions are profinite groups.

Kummer theory The Galois theory of taking n-th roots, given enough roots of unity. It includes the general theoryof quadratic extensions.

Artin–Schreier theory Covers an exceptional case of Kummer theory, in characteristic p.

Normal basis A basis in the vector space sense of L over K, on which the Galois group of L over K acts transitively.

Tensor product of fields A different foundational piece of algebra, including the compositum operation (join offields).

7.7. EXTENSIONS OF GALOIS THEORY 69

7.7 Extensions of Galois theoryInverse problem of Galois theory Given a group G, find an extension of the rational number or other field with G

as Galois group.

Differential Galois theory The subject in which symmetry groups of differential equations are studied along thelines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Liefounded the theory of Lie groups. It has not, probably, reached definitive form.

Grothendieck’s Galois theory Avery abstract approach from algebraic geometry, introduced to study the analogueof the fundamental group.

7.8 References[1] Fried & Jarden (2008) p.45

[2] Fried & Jarden (2008) p.214

[3] Serre (1992) p.19

[4] Schinzel (2000) p.298

[5] Schinzel (2000) p.5

[6] Washington, Lawrence C. (1996). Introduction to Cyclotomic fields (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0. Zbl 0966.11047.

[7] Lam (2005) p.342

[8] Fried & Jarden (2008) p.564

[9] Roman (2007) p.46

[10] Lang (2002) p.243

[11] Fried & Jarden (2008) p.28

[12] Fried & Jarden (2008) p.44

[13] Roman (2007) p.102

[14] Isaacs, I. Martin (1994). Algebra: A Graduate Course. Graduate studies in mathematics 100. American MathematicalSociety. p. 389. ISBN 0-8218-4799-6. ISSN 1065-7339.

[15] Roman (2007) p.273

[16] Cohn, P. M. (2003). Basic Algebra. Groups, Rings, and Fields. Springer-Verlag. p. 427. ISBN 1-85233-587-4. Zbl1003.00001.

[17] Lang (2002) p.228

• Adamson, Iain T. (1982). Introduction to Field Theory (2nd ed.). Cambridge University Press. ISBN 0-521-28658-1.

• Fried,Michael D.; Jarden,Moshe (2008). Field arithmetic. Ergebnisse derMathematik und ihrer Grenzgebiete.3. Folge 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.

• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67.American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.

• Lang, Serge (1997). Survey ofDiophantine Geometry. Springer-Verlag. ISBN3-540-61223-8. Zbl 0869.11051.

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

70 CHAPTER 7. GLOSSARY OF FIELD THEORY

• Roman, Steven (2007). Field Theory. Graduate Texts in Mathematics 158. Springer-Verlag. ISBN 0-387-27678-5.

• Serre, Jean-Pierre (1989). Lectures on the Mordell-Weil Theorem. Aspects of Mathematics E15. Translatedand edited by Martin Brown from notes by Michel Waldschmidt. Braunschweig etc.: Friedr. Vieweg & Sohn.Zbl 0676.14005.

• Serre, Jean-Pierre (1992). Topics in Galois Theory. Research Notes in Mathematics 1. Jones and Bartlett.ISBN 0-86720-210-6. Zbl 0746.12001.

• Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics andIts Applications 77. Cambridge: Cambridge University Press. ISBN 0-521-66225-7. Zbl 0956.12001.

Chapter 8

Glossary of game theory

Game theory is the branch of mathematics in which games are studied: that is, models describing human behaviour.This is a glossary of some terms of the subject.

8.1 Definitions of a game

8.1.1 Notational conventions

Real numbers R .

The set of players N .

Strategy space Σ =∏

i∈NΣi , where

Player i’s strategy space Σ i is the space of all possible ways in which player i can play the game.

A strategy for player i

σ i is an element of Σ i .

Complements

σ −i an element of Σ −i =∏

j∈N,j =i Σj , is a tuple of strategies for all players other than i.

Outcome space Γ is in most textbooks identical to -

Payoffs RN , describing how much gain (money, pleasure, etc.) the players are allocated by the end of the game.

8.1.2 Normal form game

A game in normal form is a function:

π :∏i∈N

Σ i → RN

Given the tuple of strategies chosen by the players, one is given an allocation of payments (given as real numbers).A further generalization can be achieved by splitting the game into a composition of two functions:

π :∏i∈N

Σ i → Γ

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72 CHAPTER 8. GLOSSARY OF GAME THEORY

the outcome function of the game (some authors call this function “the game form”), and:

ν : Γ → RN

the allocation of payoffs (or preferences) to players, for each outcome of the game.

8.1.3 Extensive form game

This is given by a tree, where at each vertex of the tree a different player has the choice of choosing an edge. Theoutcome set of an extensive form game is usually the set of tree leaves.

8.1.4 Cooperative game

A game in which players are allowed form coalitions (and to enforce coalitionary discipline). A cooperative game isgiven by stating a value for every coalition:

ν : 2P(N) → R

It is always assumed that the empty coalition gains nil. Solution concepts for cooperative games usually assume that theplayers are forming the grand coalitionN , whose value ν(N) is then divided among the players to give an allocation.

8.1.5 Simple game

A Simple game is a simplified form of a cooperative game, where the possible gain is assumed to be either '0' or '1'.A simple game is couple (N,W), whereW is the list of “winning” coalitions, capable of gaining the loot ('1'), andN is the set of players.

8.2 Glossary

Acceptable game is a game form such that for every possible preference profiles, the game has pure nash equi-libria, all of which are pareto efficient.

Allocation of goods is a function ν : Γ → RN . The allocation is a cardinal approach for determining the good(e.g. money) the players are granted under the different outcomes of the game.

Best reply the best reply to a given complement σ −i is a strategy τ i that maximizes player i's payment. Formally,we want:∀σ i ∈ Σ i π (σ i, σ −i) ≤ π (τ i, σ −i) .

Coalition is any subset of the set of players: S ⊆ N .

Condorcet winner Given a preference ν on the outcome space, an outcome a is a condorcet winner if all non-dummy players prefer a to all other outcomes.

Dictator A player is a strong dictator if he can guarantee any outcome regardless of the other players. m ∈ N is aweak dictator if he can guarantee any outcome, but his strategies for doing so might depend on the complementstrategy vector. Naturally, every strong dictator is a weak dictator. Formally:m is a Strong dictator if:∀a ∈ A, ∃σ n ∈ Σ n s.t. ∀σ −n ∈ Σ −n : Γ (σ −n, σ n) = am is aWeak dictator if:∀a ∈ A, ∀σ −n ∈ Σ −n ∃σ n ∈ Σ n s.t. Γ (σ −n, σ n) = a

8.2. GLOSSARY 73

Another way to put it is:a weak dictator is α -effective for every possible outcome.A strong dictator is β -effective for every possible outcome.A game can have no more than one strong dictator. Some games have multiple weak dictators (in rock-paper-scissorsboth players are weak dictators but none is a strong dictator).See Effectiveness. Antonym: dummy.

Dominated outcome Given a preference ν on the outcome space, we say that an outcome a is dominated byoutcome b (hence, b is the dominant strategy) if it is preferred by all players. If, in addition, some playerstrictly prefers b over a, then we say that a is strictly dominated. Formally:∀j ∈ N ν j(a) ≤ ν j(b) for domination, and∃i ∈ N s.t. ν i(a) < ν i(b) for strict domination.An outcome a is (strictly) dominated if it is (strictly) dominated by some other outcome.An outcome a is dominated for a coalition S if all players in S prefer some other outcome to a. See alsoCondorcet winner.

Dominated strategy we say that strategy is (strongly) dominated by strategy τ i if for any complement strategiestuple σ −i , player i benefits by playing τ i . Formally speaking:∀σ −i ∈ Σ −i π (σ i, σ −i) ≤ π (τ i, σ −i) and∃σ −i ∈ Σ −i s.t. π (σ i, σ −i) < π (τ i, σ −i) .A strategy σ is (strictly) dominated if it is (strictly) dominated by some other strategy.

Dummy A player i is a dummy if he has no effect on the outcome of the game. I.e. if the outcome of the game isinsensitive to player i's strategy.

Antonyms: say, veto, dictator.

Effectiveness A coalition (or a single player) S is effective for a if it can force a to be the outcome of the game. Sis α-effective if the members of S have strategies s.t. no matter what the complement of S does, the outcomewill be a.

S is β-effective if for any strategies of the complement of S, the members of S can answer with strategies that ensureoutcome a.

Finite game is a game with finitely many players, each of which has a finite set of strategies.

Grand coalition refers to the coalition containing all players. In cooperative games it is often assumed that thegrand coalition forms and the purpose of the game is to find stable imputations.

Mixed strategy for player i is a probability distribution P on Σ i . It is understood that player i chooses a strategyrandomly according to P.

Mixed Nash Equilibrium Same as Pure Nash Equilibrium, defined on the space of mixed strategies. Everyfinite game hasMixed Nash Equilibria.

Pareto efficiency An outcome a of game form π is (strongly) pareto efficient if it is undominated under allpreference profiles.

Preference profile is a function ν : Γ → RN . This is the ordinal approach at describing the outcome of thegame. The preference describes how 'pleased' the players are with the possible outcomes of the game. Seeallocation of goods.

Pure Nash Equilibrium An element σ = (σ i)i∈N of the strategy space of a game is a pure nash equilibrium pointif no player i can benefit by deviating from his strategy σ i , given that the other players are playing in σ .Formally:∀i ∈ N ∀τ i ∈ Σ i π (τ , σ −i) ≤ π (σ ) .No equilibrium point is dominated.

74 CHAPTER 8. GLOSSARY OF GAME THEORY

Say A player i has a Say if he is not a Dummy, i.e. if there is some tuple of complement strategies s.t. π (σ_i) isnot a constant function.

Antonym: Dummy.

Value A value of a game is a rationally expected outcome. There are more than a few definitions of value, de-scribing different methods of obtaining a solution to the game.

Veto A veto denotes the ability (or right) of some player to prevent a specific alternative from being the outcome ofthe game. A player who has that ability is called a veto player.

Antonym: Dummy.

Weakly acceptable game is a game that has pure nash equilibria some of which are pareto efficient.

Zero sum game is a game in which the allocation is constant over different outcomes. Formally:∀γ ∈ Γ

∑i∈N ν i(γ ) = const.

w.l.g. we can assume that constant to be zero. In a zero sum game, one player’s gain is another player’s loss.Most classical board games (e.g. chess, checkers) are zero sum.

Chapter 9

Glossary of graph theory

Graph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors usethe same word with different meanings. Some authors use different words to mean the same thing. This page attemptsto describe the majority of current usage.

9.1 Basics

A graph G consists of two types of elements, namely vertices and edges. Every edge has two endpoints in the set ofvertices, and is said to connect or join the two endpoints. An edge can thus be defined as a set of two vertices (or anordered pair, in the case of a directed graph - see Section Direction). The two endpoints of an edge are also said tobe adjacent to each other.Alternative models of graphs exist; e.g., a graph may be thought of as a Boolean binary function over the set ofvertices or as a square (0,1)-matrix.A vertex is simply drawn as a node or a dot. The vertex set of G is usually denoted by V(G), or V when there is nodanger of confusion. The order of a graph is the number of its vertices, i.e. |V(G)|.An edge (a set of two elements) is drawn as a line connecting two vertices, called endpoints or end vertices orendvertices. An edge with endvertices x and y is denoted by xy (without any symbol in between). The edge set ofG is usually denoted by E(G), or E when there is no danger of confusion. An edge xy is called incident to a vertexwhen this vertex is one of the endpoints x or y.The size of a graph is the number of its edges, i.e. |E(G)|.[1]

A loop is an edge whose endpoints are the same vertex. A link has two distinct endvertices. An edge is multiple ifthere is another edge with the same endvertices; otherwise it is simple. The multiplicity of an edge is the numberof multiple edges sharing the same end vertices; themultiplicity of a graph, the maximum multiplicity of its edges.A graph is a simple graph if it has no multiple edges or loops, a multigraph if it has multiple edges, but no loops,and amultigraph or pseudograph if it contains both multiple edges and loops (the literature is highly inconsistent).When stated without any qualification, a graph is usually assumed to be simple, except in the literature of categorytheory, where it refers to a quiver.Graphs whose edges or vertices have names or labels are known as labeled, those without as unlabeled. Graphs withlabeled vertices only are vertex-labeled, those with labeled edges only are edge-labeled. The difference between alabeled and an unlabeled graph is that the latter has no specific set of vertices or edges; it is regarded as another wayto look upon an isomorphism type of graphs. (Thus, this usage distinguishes between graphs with identifiable vertexor edge sets on the one hand, and isomorphism types or classes of graphs on the other.)(Graph labeling usually refers to the assignment of labels (usually natural numbers, usually distinct) to the edges andvertices of a graph, subject to certain rules depending on the situation. This should not be confused with a graph’smerely having distinct labels or names on the vertices.)A hyperedge is an edge that is allowed to take on any number of vertices, possibly more than 2. A graph that allowsany hyperedge is called a hypergraph. A simple graph can be considered a special case of the hypergraph, namelythe 2-uniform hypergraph. However, when stated without any qualification, an edge is always assumed to consist of

75

76 CHAPTER 9. GLOSSARY OF GRAPH THEORY

In this pseudograph the blue edges are loops and the red edges are multiple edges of multiplicity 2 and 3. The multiplicity of thegraph is 3.

at most 2 vertices, and a graph is never confused with a hypergraph.A non-edge (or anti-edge) is an edge that is not present in the graph. More formally, for two vertices u and v , {u, v}is a non-edge in a graphG whenever {u, v} is not an edge inG . This means that there is either no edge between thetwo vertices or (for directed graphs) at most one of (u, v) and (v, u) from v is an arc in G.Occasionally the term cotriangle or anti-triangle is used for a set of three vertices none of which are connected.The complement G of a graph G is a graph with the same vertex set as G but with an edge set such that xy is an edgein G if and only if xy is not an edge in G.An edgeless graph or empty graph or null graph is a graph with zero or more vertices, but no edges. The emptygraph or null graph may also be the graph with no vertices and no edges. If it is a graph with no edges and anynumber n of vertices, it may be called the null graph on n vertices. (There is no consistency at all in the literature.)A graph is infinite if it has infinitely many vertices or edges or both; otherwise the graph is finite. An infinite graphwhere every vertex has finite degree is called locally finite. When stated without any qualification, a graph is usuallyassumed to be finite. See also continuous graph.Two graphs G and H are said to be isomorphic, denoted by G ~ H, if there is a one-to-one correspondence, calledan isomorphism, between the vertices of the graph such that two vertices are adjacent in G if and only if their

9.1. BASICS 77

1

23

54

6

A labeled simple graph with vertex set V = {1, 2, 3, 4, 5, 6} and edge set E = {{1,2}, {1,5}, {2,3}, {2,5}, {3,4}, {4,5}, {4,6}}.

corresponding vertices are adjacent in H. Likewise, a graph G is said to be homomorphic to a graph H if thereis a mapping, called a homomorphism, from V(G) to V(H) such that if two vertices are adjacent in G then theircorresponding vertices are adjacent in H.

9.1.1 Subgraphs

A subgraph, H, of a graph, G, is a graph whose vertices are a subset of the vertex set of G, and whose edges are asubset of the edge set of G. In reverse, a supergraph of a graph G is a graph of which G is a subgraph. A graph, G,contains a graph, H, if H is a subgraph of, or is isomorphic to G.A subgraph, H, spans a graph, G, and is a spanning subgraph, or factor of G, if it has the same vertex set as G.A subgraph, H, of a graph, G, is said to be induced (or full) if, for every pair of vertices x and y of H, xy is an edgeof H if and only if xy is an edge of G. In other words, H is an induced subgraph of G if it has exactly the edges thatappear in G over the same vertex set. If the vertex set of H is the subset S of V(G), then H can be written as G[S]and is said to be induced by S.A graph, G, is minimal with some property, P, provided that G has property P and no proper subgraph of G hasproperty P. In this definition, the term subgraph is usually understood to mean induced subgraph. The notion ofmaximality is defined dually: G is maximal with P provided that P(G) and G has no proper supergraph H such thatP(H).A graph that does not contain H as an induced subgraph is said to be H-free, and more generally if F is a family ofgraphs then the graphs that do not contain any induced subgraph isomorphic to a member of F are called F -free.[2]For example the triangle-free graphs are the graphs that do not have a triangle graph as an induced subgraph. Manyimportant classes of graphs can be defined by sets of forbidden subgraphs, the graphs that are not in the class and areminimal with respect to subgraphs, induced subgraphs, or graph minors.A universal graph in a class K of graphs is a simple graph in which every element in K can be embedded as asubgraph.

78 CHAPTER 9. GLOSSARY OF GRAPH THEORY

9.1.2 Walks

A walk (sometimes called a chain)[3] is a sequence of alternating vertices and edges, beginning and ending withvertices, where each edge’s endpoints are the preceding and following vertices in the sequence. A walk is closed ifits first and last vertices are the same, and open if they are different.The length l of a walk is the number of edges that it uses. For an open walk, l = n–1, where n is the number ofvertices visited (a vertex is counted each time it is visited). For a closed walk, l = n (the start/end vertex is listedtwice, but is not counted twice). In the example labeled simple graph, (1, 2, 5, 1, 2, 3) is an open walk with length 5,and (4, 5, 2, 1, 5, 4) is a closed walk of length 5.A trail is a walk in which all the edges are distinct. A closed trail is sometimes called a tour or circuit, but theseusages are not universal, and the latter is often reserved for a regular subgraph of degree two.

A directed tour. This is not a simple cycle, since the blue vertices are used twice.

Traditionally, a path referred to what is now usually known as an open walk. Nowadays, when stated without anyqualification, a path is usually understood to be simple, meaning that no vertices (and thus no edges) are repeated.(The term chain has also sometimes been used to mean a path in the modern sense.) In the example labeled simplegraph, (5, 2, 1) is a path of length 2. The closed equivalent to this type of walk, a walk that starts and ends at thesame vertex but otherwise has no repeated vertices or edges, is called a cycle or closed path. Like path, this termtraditionally referred to any closed walk, but now is usually understood to be simple by definition. In the examplelabeled simple graph, (1, 5, 2, 1) is a cycle of length 3. (A cycle, unlike a path, is not allowed to have length 0.)Paths and cycles of n vertices are often denoted by Pn and Cn, respectively. (Some authors use as subscript the lengthinstead of the number of vertices, however.)

9.1. BASICS 79

C1 is a loop, C2 is a digon (a pair of parallel undirected edges in a multigraph, or a pair of antiparallel edges in adirected graph), and C3 is called a triangle.A cycle that has odd length is an odd cycle; otherwise it is an even cycle. One theorem is that a graph is bipartite ifand only if it contains no odd cycles. (See complete bipartite graph.)A graph is acyclic if it contains no cycles; unicyclic if it contains exactly one cycle; and pancyclic if it contains cyclesof every possible length (from 3 to the order of the graph).A wheel graph is a graph with n vertices (n ≥ 4), formed by connecting a single vertex to all vertices of C -₁.The girth of a graph is the length of a shortest (simple) cycle in the graph; and the circumference, the length of alongest (simple) cycle. The girth and circumference of an acyclic graph are defined to be infinity, ∞.A path or cycle is Hamiltonian (or spanning) if it uses all vertices exactly once. A graph that contains a Hamil-tonian path is traceable; and one that contains a Hamiltonian path for any given pair of (distinct) end vertices is aHamiltonian connected graph. A graph that contains a Hamiltonian cycle is a Hamiltonian graph.A trail or circuit (or cycle) is Eulerian if it uses all edges precisely once. A graph that contains an Eulerian trail istraversable. A graph that contains an Eulerian circuit is an Eulerian graph.Two paths are internally disjoint (some people call it independent) if they do not have any vertex in common, exceptthe first and last ones.A theta graph is the union of three internally disjoint (simple) paths that have the same two distinct end vertices.[4]Some authors have used the term theta0 graph to refer to a graph that has seven vertices and eight edges that canbe drawn as the perimeter and one diameter of a regular hexagon. (The seventh vertex splits the diameter into twoedges.) The smallest, excluding multigraphs, topological minor of a theta0 graph consists of a square plus one of itsdiagonals.

9.1.3 Trees

A tree is a connected acyclic simple graph. For directed graphs, each vertex has at most one incoming edge. A vertexof degree 1 is called a leaf, or pendant vertex. An edge incident to a leaf is a leaf edge, or pendant edge. (Somepeople define a leaf edge as a leaf and then define a leaf vertex on top of it. These two sets of definitions are oftenused interchangeably.) A non-leaf vertex is an internal vertex. Sometimes, one vertex of the tree is distinguished,and called the root; in this case, the tree is called rooted. Rooted trees are often treated as directed acyclic graphswith the edges pointing away from the root.A subtree of the tree T is a connected subgraph of T.A forest is an acyclic simple graph. For directed graphs, each vertex has at most one incoming edge. (That is, a treewith the connectivity requirement removed; a graph containing multiple disconnected trees.)A subforest of the forest F is a subgraph of F.A spanning tree is a spanning subgraph that is a tree. Every graph has a spanning forest. But only a connected graphhas a spanning tree.A special kind of tree called a star is K₁,k. An induced star with 3 edges is a claw.A caterpillar is a tree in which all non-leaf nodes form a single path.A k-ary tree is a rooted tree in which every internal vertex has no more than k children. A 1-ary tree is just a path.A 2-ary tree is also called a binary tree.

9.1.4 Cliques

The complete graph Kn of order n is a simple graph with n vertices in which every vertex is adjacent to every other.The pentagon-shaped graph to the right is complete. The complete graph on n vertices is often denoted by Kn. It hasn(n−1)/2 edges (corresponding to all possible choices of pairs of vertices).A clique in a graph is a set of pairwise adjacent vertices. Since any subgraph induced by a clique is a completesubgraph, the two terms and their notations are usually used interchangeably. A k-clique is a clique of order k. In theexample labeled simple graph above, vertices 1, 2 and 5 form a 3-clique, or a triangle. A maximal clique is a cliquethat is not a subset of any other clique (some authors reserve the term clique for maximal cliques).

80 CHAPTER 9. GLOSSARY OF GRAPH THEORY

4

1 2

3

5

6A labeled tree with 6 vertices and 5 edges. Nodes 1, 2, 3, and 6 are leaves, while 4 and 5 are internal vertices.

The clique number ω(G) of a graph G is the order of a largest clique in G.

9.1.5 Strongly connected component

A related but weaker concept is that of a strongly connected component. Informally, a strongly connected componentof a directed graph is a subgraph where all nodes in the subgraph are reachable by all other nodes in the subgraph.Reachability between nodes is established by the existence of a path between every pair of nodes in the component.A directed graph can be partitioned into strongly connected components by running the depth-first search (DFS)algorithm twice: first, on the graph itself and next on the transpose graph in decreasing order of the finishing timesof the first DFS. Given a directed graph G, the transpose GT is the graph G with all the edge directions reversed.

9.1. BASICS 81

K5, a complete graph. If a subgraph looks like this, the vertices in that subgraph form a clique of size 5.

9.1.6 Hypercubes

A hypercube graphQn is a regular graph with 2n vertices, 2n−1n edges, and n edges touching each vertex. It can beobtained as the one-dimensional skeleton of the geometric hypercube.

9.1.7 Knots

A knot in a directed graph is a collection of vertices and edges with the property that every vertex in the knot hasoutgoing edges, and all outgoing edges from vertices in the knot terminate at other vertices in the knot. Thus it isimpossible to leave the knot while following the directions of the edges.

9.1.8 Minors

A graphG2 is a minor of a graphG1 ifG2 can be obtained fromG1 by a sequence of edge deletions, isolated vertexdeletions, and edge contractions that merge the two endpoints of an edge as a single vertex. A related concept istopological minors, where G2 = (V2, E2) is a topological minor of G1 = (V1, E1) if there is an injection from V2

to V1 such that every edge in E2 corresponds to a path (disjoint from all other such paths) in G1 .

82 CHAPTER 9. GLOSSARY OF GRAPH THEORY

9.1.9 Embedding

An embeddingG2 = (V2, E2) ofG1 = (V1, E1) is an injection from V2 to V1 such that every edge inE2 correspondsto a path in G1 .[5]

9.2 Adjacency and degree

In graph theory, degree, especially that of a vertex, is usually a measure of immediate adjacency.An edge connects two vertices; these two vertices are said to be incident to that edge, or, equivalently, that edgeincident to those two vertices. All degree-related concepts have to do with adjacency or incidence.The degree, or valency, dG(v) of a vertex v in a graph G is the number of edges incident to v, with loops beingcounted twice. A vertex of degree 0 is an isolated vertex. A vertex of degree 1 is a leaf. In the example labeledsimple graph, vertices 1 and 3 have a degree of 2, vertices 2, 4 and 5 have a degree of 3, and vertex 6 has a degree of1. If E is finite, then the total sum of vertex degrees is equal to twice the number of edges.The total degree of a graph is the sum of the degrees of all its vertices. Thus, for a graph without loops, it is equal tothe number of incidences between vertices and edges. The handshaking lemma states that the total degree is alwaysequal to two times the number of edges, loops included. This means that for a simple graph with 3 vertices with eachvertex having a degree of two (i.e. a triangle) the total degree would be six (e.g. 3 x 2 = 6).A degree sequence is a list of degrees of a graph in non-increasing order (e.g. d1 ≥ d2 ≥ … ≥ dn). A sequence ofnon-increasing integers is realizable if it is a degree sequence of some graph.Two vertices u and v are called adjacent if an edge exists between them. We denote this by u ~ v or u ↓ v. In theabove graph, vertices 1 and 2 are adjacent, but vertices 2 and 4 are not. The set of neighbors of v, that is, verticesadjacent to v not including v itself, forms an induced subgraph called the (open) neighborhood of v and denotedNG(v). When v is also included, it is called a closed neighborhood and denoted by NG[v]. When stated withoutany qualification, a neighborhood is assumed to be open. The subscript G is usually dropped when there is no dangerof confusion; the same neighborhood notation may also be used to refer to sets of adjacent vertices rather than thecorresponding induced subgraphs. In the example labeled simple graph, vertex 1 has two neighbors: vertices 2 and5. For a simple graph, the number of neighbors that a vertex has coincides with its degree.A dominating set of a graph is a vertex subset whose closed neighborhood includes all vertices of the graph. A vertexv dominates another vertex u if there is an edge from v to u. A vertex subset V dominates another vertex subsetU if every vertex in U is adjacent to some vertex in V. The minimum size of a dominating set is the dominationnumber γ(G).In computers, a finite, directed or undirected graph (with n vertices, say) is often represented by its adjacencymatrix:an n-by-n matrix whose entry in row i and column j gives the number of edges from the i-th to the j-th vertex.Spectral graph theory studies relationships between the properties of a graph and its adjacency matrix or othermatrices associated with the graph.The maximum degree Δ(G) of a graph G is the largest degree over all vertices; the minimum degree δ(G), thesmallest.A graph in which every vertex has the same degree is regular. It is k-regular if every vertex has degree k. A 0-regular graph is an independent set. A 1-regular graph is a matching. A 2-regular graph is a vertex disjoint union ofcycles. A 3-regular graph is said to be cubic, or trivalent.A k-factor is a k-regular spanning subgraph. A 1-factor is a perfect matching. A partition of edges of a graph intok-factors is called a k-factorization. A k-factorable graph is a graph that admits a k-factorization.A graph is biregular if it has unequal maximum andminimum degrees and every vertex has one of those two degrees.A strongly regular graph is a regular graph such that any adjacent vertices have the same number of commonneighbors as other adjacent pairs and that any nonadjacent vertices have the same number of common neighbors asother nonadjacent pairs.

9.3. COMPLEXITY 83

9.2.1 Independence

In graph theory, the word independent usually carries the connotation of pairwise disjoint or mutually nonadjacent.In this sense, independence is a form of immediate nonadjacency. An isolated vertex is a vertex not incident to anyedges. An independent set, or coclique, or stable set, is a set of vertices of which no pair is adjacent. Since the graphinduced by any independent set is an empty graph, the two terms are usually used interchangeably. In the examplelabeled simple graph at the top of this page, vertices 1, 3, and 6 form an independent set; and 2 and 4 form anotherone.Two subgraphs are edge disjoint if they have no edges in common. Similarly, two subgraphs are vertex disjoint ifthey have no vertices (and thus, also no edges) in common. Unless specified otherwise, a set of disjoint subgraphsare assumed to be pairwise vertex disjoint.The independence number α(G) of a graph G is the size of the largest independent set of G.A graph can be decomposed into independent sets in the sense that the entire vertex set of the graph can be partitionedinto pairwise disjoint independent subsets. Such independent subsets are called partite sets, or simply parts.A graph that can be decomposed into two partite sets bipartite; three sets, tripartite; k sets, k-partite; and anunknown number of sets,multipartite. An 1-partite graph is the same as an independent set, or an empty graph. A2-partite graph is the same as a bipartite graph. A graph that can be decomposed into k partite sets is also said to bek-colourable.A complete multipartite graph is a graph in which vertices are adjacent if and only if they belong to different partitesets. A complete bipartite graph is also referred to as a biclique; if its partite sets contain n andm vertices, respectively,then the graph is denoted Kn,m.A k-partite graph is semiregular if each of its partite sets has a uniform degree; equipartite if each partite set hasthe same size; and balanced k-partite if each partite set differs in size by at most 1 with any other.Thematching number α′(G) of a graph G is the size of a largestmatching, or pairwise vertex disjoint edges, of G.A spanning matching, also called a perfect matching is a matching that covers all vertices of a graph.

9.3 Complexity

Complexity of a graph denotes the quantity of information that a graph contained, and can be measured in severalways. For example, by counting the number of its spanning trees, or the value of a certain formula involving thenumber of vertices, edges, and proper paths in a graph. [6]

9.4 Connectivity

Connectivity extends the concept of adjacency and is essentially a form (and measure) of concatenated adjacency.If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is said to be connected;otherwise, the graph is disconnected. A graph is totally disconnected if there is no path connecting any pair ofvertices. This is just another name to describe an empty graph or independent set.A cut vertex, or articulation point, is a vertex whose removal disconnects the remaining subgraph. A cut set, orvertex cut or separating set, is a set of vertices whose removal disconnects the remaining subgraph. A bridge is ananalogous edge (see below).If it is always possible to establish a path from any vertex to every other even after removing any k - 1 vertices,then the graph is said to be k-vertex-connected or k-connected. Note that a graph is k-connected if and only if itcontains k internally disjoint paths between any two vertices. The example labeled simple graph above is connected(and therefore 1-connected), but not 2-connected. The vertex connectivity or connectivity κ(G) of a graph G is theminimum number of vertices that need to be removed to disconnect G. The complete graph Kn has connectivity n -1 for n > 1; and a disconnected graph has connectivity 0.In network theory, a giant component is a connected subgraph that contains a majority of the entire graph’s nodes.A bridge, or cut edge or isthmus, is an edge whose removal disconnects a graph. (For example, all the edges in a treeare bridges.) A cut vertex is an analogous vertex (see above). A disconnecting set is a set of edges whose removal

84 CHAPTER 9. GLOSSARY OF GRAPH THEORY

increases the number of components. An edge cut is the set of all edges which have one vertex in some proper vertexsubset S and the other vertex in V(G)\S. Edges of K3 form a disconnecting set but not an edge cut. Any two edgesof K3 form a minimal disconnecting set as well as an edge cut. An edge cut is necessarily a disconnecting set; and aminimal disconnecting set of a nonempty graph is necessarily an edge cut. A bond is a minimal (but not necessarilyminimum), nonempty set of edges whose removal disconnects a graph.A graph is k-edge-connected if any subgraph formed by removing any k - 1 edges is still connected. The edgeconnectivity κ'(G) of a graph G is the minimum number of edges needed to disconnect G. One well-known result isthat κ(G) ≤ κ'(G) ≤ δ(G).A component is a maximally connected subgraph. A block is either a maximally 2-connected subgraph, a bridge(together with its vertices), or an isolated vertex. A biconnected component is a 2-connected component.An articulation point (also known as a separating vertex) of a graph is a vertex whose removal from the graphincreases its number of connected components. A biconnected component can be defined as a subgraph induced bya maximal set of nodes that has no separating vertex.

9.5 Distance

The distance dG(u, v) between two (not necessarily distinct) vertices u and v in a graph G is the length of a shortestpath (also called a graph geodesic) between them. The subscript G is usually dropped when there is no danger ofconfusion. When u and v are identical, their distance is 0. When u and v are unreachable from each other, theirdistance is defined to be infinity ∞.The eccentricity εG(v) of a vertex v in a graph G is the maximum distance from v to any other vertex. The diameterdiam(G) of a graph G is the maximum eccentricity over all vertices in a graph; and the radius rad(G), the minimum.When there are two components in G, diam(G) and rad(G) defined to be infinity ∞. Trivially, diam(G) ≤ 2 rad(G).Vertices with maximum eccentricity are called peripheral vertices. Vertices of minimum eccentricity form thecenter. A tree has at most two center vertices.TheWiener index of a vertex v in a graph G, denoted by WG(v) is the sum of distances between v and all others.TheWiener index of a graph G, denoted byW(G), is the sum of distances over all pairs of vertices. An undirectedgraph’sWiener polynomial is defined to be Σ qd(u,v) over all unordered pairs of vertices u and v. Wiener index andWiener polynomial are of particular interest to mathematical chemists.The k-th power Gk of a graph G is a supergraph formed by adding an edge between all pairs of vertices of G withdistance at most k. A second power of a graph is also called a square.A k-spanner is a spanning subgraph, S, in which every two vertices are at most k times as far apart on S than on G.The number k is the dilation. k-spanner is used for studying geometric network optimization.

9.6 Genus

A crossing is a pair of intersecting edges. A graph is embeddable on a surface if its vertices and edges can bearranged on it without any crossing. The genus of a graph is the lowest genus of any surface on which the graph canembed.A planar graph is one which can be drawn on the (Euclidean) plane without any crossing; and a plane graph, onewhich is drawn in such fashion. In other words, a planar graph is a graph of genus 0. The example labeled simplegraph is planar; the complete graph on n vertices, for n> 4, is not planar. Also, a tree is necessarily a planar graph.When a graph is drawn without any crossing, any cycle that surrounds a region without any edges reaching from thecycle into the region forms a face. Two faces on a plane graph are adjacent if they share a common edge. A dual,or planar dual when the context needs to be clarified, G* of a plane graph G is a graph whose vertices represent thefaces, including any outerface, of G and are adjacent in G* if and only if their corresponding faces are adjacent in G.The dual of a planar graph is always a planar pseudograph (e.g. consider the dual of a triangle). In the familiar caseof a 3-connected simple planar graph G (isomorphic to a convex polyhedron P), the dual G* is also a 3-connectedsimple planar graph (and isomorphic to the dual polyhedron P*).Furthermore, since we can establish a sense of “inside” and “outside” on a plane, we can identify an “outermost”region that contains the entire graph if the graph does not cover the entire plane. Such outermost region is called

9.7. WEIGHTED GRAPHS AND NETWORKS 85

an outer face. An outerplanar graph is one which can be drawn in the planar fashion such that its vertices are alladjacent to the outer face; and an outerplane graph, one which is drawn in such fashion.The minimum number of crossings that must appear when a graph is drawn on a plane is called the crossing number.The minimum number of planar graphs needed to cover a graph is the thickness of the graph.

9.7 Weighted graphs and networks

A weighted graph associates a label (weight) with every edge in the graph. Weights are usually real numbers. Theymay be restricted to rational numbers or integers. Certain algorithms require further restrictions on weights; forinstance, Dijkstra’s algorithm works properly only for positive weights. The weight of a path or the weight of atree in a weighted graph is the sum of the weights of the selected edges. Sometimes a non-edge (a vertex pair withno connecting edge) is indicated by labeling it with a special weight representing infinity. Sometimes the word costis used instead of weight. When stated without any qualification, a graph is always assumed to be unweighted. Insome writing on graph theory the term network is a synonym for a weighted graph. A network may be directed orundirected, it may contain special vertices (nodes), such as source or sink. The classical network problems include:

• minimum cost spanning tree,

• shortest paths,

• maximal flow (and the max-flow min-cut theorem)

9.8 Direction

Main article: Digraph (mathematics)

A directed arc, or directed edge, is an ordered pair of endvertices that can be represented graphically as an arrowdrawn between the endvertices. In such an ordered pair the first vertex is called the initial vertex or tail; the secondone is called the terminal vertex or head (because it appears at the arrow head). An undirected edge disregardsany sense of direction and treats both endvertices interchangeably. A loop in a digraph, however, keeps a sense ofdirection and treats both head and tail identically. A set of arcs aremultiple, or parallel, if they share the same headand the same tail. A pair of arcs are anti-parallel if one’s head/tail is the other’s tail/head. A digraph, or directedgraph, or oriented graph, is analogous to an undirected graph except that it contains only arcs. Amixed graphmaycontain both directed and undirected edges; it generalizes both directed and undirected graphs. When stated withoutany qualification, a graph is almost always assumed to be undirected.A digraph is called simple if it has no loops and at most one arc between any pair of vertices. When stated withoutany qualification, a digraph is usually assumed to be simple. A quiver is a directed graph which is specifically allowed,but not required, to have loops and more than one arc between any pair of vertices.In a digraph Γ, we distinguish the out degree dΓ+(v), the number of edges leaving a vertex v, and the in degreedΓ−(v), the number of edges entering a vertex v. If the graph is oriented, the degree dΓ(v) of a vertex v is equal to thesum of its out- and in- degrees. When the context is clear, the subscript Γ can be dropped. Maximum and minimumout degrees are denoted by Δ+(Γ) and δ+(Γ); and maximum and minimum in degrees, Δ−(Γ) and δ−(Γ).An out-neighborhood, or successor set, N+Γ(v) of a vertex v is the set of heads of arcs going from v. Likewise, anin-neighborhood, or predecessor set, N−Γ(v) of a vertex v is the set of tails of arcs going into v.A source is a vertex with 0 in-degree; and a sink, 0 out-degree.A vertex v dominates another vertex u if there is an arc from v to u. A vertex subset S is out-dominating if everyvertex not in S is dominated by some vertex in S; and in-dominating if every vertex in S is dominated by some vertexnot in S.A kernel in a (possibly directed) graph G is an independent set S such that every vertex in V(G) \ S dominatessome vertex in S. In undirected graphs, kernels are maximal independent sets.[7] A digraph is kernel perfect if everyinduced sub-digraph has a kernel.[8]

An Eulerian digraph is a digraph with equal in- and out-degrees at every vertex.

86 CHAPTER 9. GLOSSARY OF GRAPH THEORY

The zweieck of an undirected edge e = (u, v) is the pair of diedges (u, v) and (v, u)which form the simple dicircuit.An orientation is an assignment of directions to the edges of an undirected or partially directed graph. Whenstated without any qualification, it is usually assumed that all undirected edges are replaced by a directed one in anorientation. Also, the underlying graph is usually assumed to be undirected and simple.A tournament is a digraph in which each pair of vertices is connected by exactly one arc. In other words, it is anoriented complete graph.A directed path, or just a path when the context is clear, is an oriented simple path such that all arcs go the samedirection, meaning all internal vertices have in- and out-degrees 1. A vertex v is reachable from another vertex u ifthere is a directed path that starts from u and ends at v. Note that in general the condition that u is reachable from vdoes not imply that v is also reachable from u.If v is reachable from u, then u is a predecessor of v and v is a successor of u. If there is an arc from u to v, then uis a direct predecessor of v, and v is a direct successor of u.A digraph is strongly connected if every vertex is reachable from every other following the directions of the arcs.On the contrary, a digraph is weakly connected if its underlying undirected graph is connected. A weakly connectedgraph can be thought of as a digraph in which every vertex is “reachable” from every other but not necessarily followingthe directions of the arcs. A strong orientation is an orientation that produces a strongly connected digraph.A directed cycle, or just a cycle when the context is clear, is an oriented simple cycle such that all arcs go the samedirection, meaning all vertices have in- and out-degrees 1. A digraph is acyclic if it does not contain any directedcycle. A finite, acyclic digraph with no isolated vertices necessarily contains at least one source and at least one sink.An arborescence, or out-tree or branching, is an oriented tree in which all vertices are reachable from a single vertex.Likewise, an in-tree is an oriented tree in which a single vertex is reachable from every other one.

9.8.1 Directed acyclic graphs

Main article: directed acyclic graph

The partial order structure of directed acyclic graphs (or DAGs) gives them their own terminology.If there is a directed edge from u to v, then we say u is a parent of v and v is a child of u. If there is a directed pathfrom u to v, we say u is an ancestor of v and v is a descendant of u.The moral graph of a DAG is the undirected graph created by adding an (undirected) edge between all parents ofthe same node (sometimes called marrying), and then replacing all directed edges by undirected edges. A DAG isperfect if, for each node, the set of parents is complete (i.e. no new edges need to be added when forming the moralgraph).

9.9 Colouring

Main article: Graph colouringVertices in graphs can be given colours to identify or label them. Although they may actually be rendered in diagramsin different colours, working mathematicians generally pencil in numbers or letters (usually numbers) to represent thecolours.Given a graph G (V,E) a k-colouring of G is a map ϕ : V → {1, ..., k} with the property that (u, v) ∈ E ⇒ ϕ(u) ≠ϕ(v) - in other words, every vertex is assigned a colour with the condition that adjacent vertices cannot be assignedthe same colour.The chromatic number χ(G) is the smallest k for which G has a k-colouring.Given a graph and a colouring, the colour classes of the graph are the sets of vertices given the same colour.A graph is called k-critical if its chromatic number is k but all of its proper subgraphs have chromatic number lessthan k. An odd cycle is 3-critical, and the complete graph on k vertices is k-critical.

9.10. VARIOUS 87

This graph is an example of a 4-critical graph. Its chromatic number is 4 but all of its proper subgraphs have a chromatic numberless than 4. This graph is also planar

9.10 Various

A graph invariant is a property of a graphG, usually a number or a polynomial, that depends only on the isomorphismclass of G. Examples are the order, genus, chromatic number, and chromatic polynomial of a graph.

9.11 See also• Graph (mathematics)

• List of graph theory topics

9.12 References[1] Harris, John M. (2000). Combinatorics and Graph Theory. New York: Springer-Verlag. p. 5. ISBN 0-387-98736-3.

[2] Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), “Chapter 7: Forbidden Subgraph”, Graph Classes: A Survey,SIAM Monographs on Discrete Mathematics and Applications, pp. 105–121, ISBN 0-89871-432-X.

[3] Encyclopedia Britannica online

[4] Mitchem, John (1969), “Hypo-properties in graphs”, The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ.,Kalamazoo, Mich., 1968), Springer, pp. 223–230, doi:10.1007/BFb0060121, MR 0253932; Bondy, J. A. (1972), “The“graph theory” of the Greek alphabet”, Graph theory and applications (Proc. Conf., Western Michigan Univ., Kalamazoo,Mich., 1972; dedicated to the memory of J. W. T. Youngs), Lecture Notes in Mathematics 303, Springer, pp. 43–54,doi:10.1007/BFb0067356, MR 0335362.

[5] Rosenberg, Arnold L. and Heath, Lenwood S. (2001). Graph separators with applications. (1st ed.). Kluwer. ISBN978-0-306-46464-5.

88 CHAPTER 9. GLOSSARY OF GRAPH THEORY

[6] Neel, David L. (2006), “The linear complexity of a graph”, The electronic journal of combinatorics

[7] Bondy, J.A., Murty, U.S.R., Graph Theory, p. 298

[8] Béla Bollobás, Modern Graph theory, p. 298

• Bollobás, Béla (1998). Modern Graph Theory. Graduate Texts in Mathematics 184. New York: Springer-Verlag. ISBN 0-387-98488-7. Zbl 0902.05016.. [Packed with advanced topics followed by a historicaloverview at the end of each chapter.]

• Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy P. (1999). Graph classes: a survey. SIAM Monographson Discrete Mathematics. and Applications 3. Philadelphia, PA: Society for Industrial and Applied Mathe-matics. ISBN 978-0-898714-32-6. Zbl 0919.05001.

• Diestel, Reinhard (2010). Graph Theory. Graduate Texts in Mathematics 173 (4th ed.). Springer-Verlag.ISBN 978-3-642-14278-9. Zbl 1204.05001. [Standard textbook, most basic material and some deeper results,exercises of various difficulty and notes at the end of each chapter; known for being quasi error-free.]

• West, Douglas B. (2001). Introduction to Graph Theory (2ed). Upper Saddle River: Prentice Hall. ISBN0-13-014400-2. [Tons of illustrations, references, and exercises. The most complete introductory guide to thesubject.]

• Weisstein, Eric W., “Graph”, MathWorld.

• Zaslavsky, Thomas. Glossary of signed and gain graphs and allied areas. Electronic Journal of Combinatorics,Dynamic Surveys in Combinatorics, # DS 9. http://www.combinatorics.org/issue/view/Surveys

Chapter 10

Glossary of invariant theory

This page is a glossary of terms in invariant theory. For descriptions of particular invariant rings, see invariants ofa binary form, symmetric polynomials. For geometric terms used in invariant theory see the glossary of classicalalgebraic geometry. Definitions of many terms used in invariant theory can be found in (Sylvester 1853), (Cayley1860), (Burnside & Panton 1881), (Salmon 1885), (Elliot 1895), (Grace & Young 1903), (Glenn 1915), (Dolgachev2012), and the index to the fourth volume of Sylvester’s collected works includes many of the terms invented by him.Contents :

• Conventions

• !$@

• A

• B

• C

• D

• E

• F

• G

• H

• I

• J

• K

• L

• M

• N

• O

• P

• Q

• R

• S

89

90 CHAPTER 10. GLOSSARY OF INVARIANT THEORY

• T

• U

• V

• W

• XYZ

•

•

• See also

• References

10.1 Conventions-an Nouns ending in -an are often invariants named after people, as in Cayleyan, Hessian, Jacobian, Steinerian.

-ant Nouns ending in -ant are often invariants, as in determinant, covariant, and so on.

-ary Adjectives ending in -ary often refer to the number of variables of a form, as in unary, binary, ternary, quater-nary, quinary, senary, septenary, octonary, nonary, denary.

-ic Adjectives or nouns ending in -ic often refer to the degree of a form, as in linear or monic, quadric or quadratic,cubic, quartic or biquadratic, quintic, sextic, septic or septimic, octic or octavic, nonic, decic or decimic,undecic or undecimic, duodecic or duodecimic, and so on.

10.2 !$@(a0, a1, ..., an)(x,y)n Short for the form (n

0)a0xn + (n1)a1xn–1y+ ... + (nn)anyn. When the first ) has a circumflex or arrow on top of it, this means that the binomial coefficients areomitted. The parentheses are sometimes overlapped: (a0, . . . , an)(x, y)n

[] See Sylvester (1853, Glossary p. 543–548)

(αβγ...) The determinant of the matrix with entries αi, βi, γi,... For example, (αβ) means α1β2 – α2β1.

10.3 Aabsolute 1. The absolute invariant is essentially the j-invariant of an elliptic curve.

2. An absolute invariant is something fixed by a group action, in other words a (relative) invariant (something thattransforms according to a character) where the character is trivial.

allotrious See Sylvester (1853, Glossary p. 543–548), Archaic.

alternant 1. An archaic term for the commutator AB–BA of two operators A and B. (Elliott 1895, p.144)

2. An alternant matrix is a matrix such that the entries of each column are given by some fixed function of a variable.

annihilator An annihilator is a differential operator representing an element of a Lie algebra, so that invariants of agroup are killed by the annihilators. (Elliott 1895, p.108)

anti-invariant A relative invariant transforming according to a character of order 2 of a group such as the symmetricgroup.

10.4. B 91

anti-seminvariant (Elliott 1895, p.126)

apocopated See Sylvester (1853, Glossary p. 543–548). Archaic.

Arf invariant Main article: Arf invariantAn invariant of quadratic forms over a field of order 2.

Aronhold invariant One of the two generators of degrees 4 and 6 of the ring of invariants of ternary cubic forms.(Dolgachev 2012, 3.1.1)

asyzygetic Linearly independent.

10.4 B

Bezoutiant Main article: BezoutiantA symmetric square matrix associated to two binary forms.

Bezoutic See Sylvester (1853, Glossary p. 543–548). Archaic.

Bezoutiod See Sylvester (1853, Glossary p. 543–548). Archaic.

bidegree An ordered pair of integers, giving the degrees of a form relative to two sets of variables.

biform A polynomial homogeneous in each of two sets of variables. In other words an element of SmV×SnW, usuallyconsidered as a representation of GLV×GLW.

binary Depending on 2 variables. Same as bivariate.

biquadratic Same as quartic, meaning degree 4.

biternary A biternary form is one in 6 variables, 3 transforming according to the fundamental representation of SL3

and 3 transforming according to its dual.

bivariate Depending on 2 variables. Same as binary.

Boolian invariant An invariant for the orthogonal group. (Elliott 1895, p.344)

bordered Hessian An alternative name for the reciprocant

bracket An invariant given by either the pairing of a vector and a vector in the dual space, or the determinant of amatrix form by n vectors of an n-dimensional space (in other words their exterior product in the top exteriorpower).

Brioschi covariant This is a degree 12 order 9 covariant of ternary cubic forms, introduced by Brioschi (1863).(Dolgachev 2012, 3.4.3)

10.5 C

canonical form A particularly simple representation of a form, such as a sum of powers of linear forms, or withmany zero coefficients. For example, the canonical form of a binary form of degree 2m+1 is a sum of m+1powers of linear forms.

canonisant

canonizant Main article: canonizantA covariant of a form, given by the catalecticant of the penultimate emanant. It is related to the canonical formof a form. For example, the canonizant of a binary form of degree 2n–1 has degree n and order n. (Elliott1895, p.21)

catalecticant Main article: catalecticantAn invariant vanishing on forms that are the sum of an unusually small number of powers of linear forms.

92 CHAPTER 10. GLOSSARY OF INVARIANT THEORY

Cayley Ω process Main article: Cayley’s Ω processA certain differential operator used for constructing invariants.

Cayleyan Main article: CayleyanA contravariant.

characteristic See Sylvester (1853, Glossary p. 543–548)

class The class of a contravariant or concomitant is its degree in the covariant variables. See also degree and order.

Clebsch invariant (Dolgachev 2012, p.283)

co-Bezoutiant See Sylvester (1853, Glossary p. 543–548). Archaic.

cogredient Transforming according to the natural representation of a linear group. (Elliott 1895, p.55)

combinant A joint relative invariant of several forms of the same degree, that is unchanged if a multiple of one ofthe forms is added to another. Essentially a relative invariant of a product of two general linear groups. (Elliott1895, p.340) Sylvester (1853, Glossary p. 543–548) (Salmon 1885, p.161)

combinative Related to invariants of a product of groups. For example a combinative covariant is a covariant of aproduct of two groups.

commutant A generalization of the determinant to arrays of dimension greater than 2. (Cayley 1860)

complete A complete system of invariants is a set of generators for the ring of invariants.

concomitant A relative invariant of GL(V) acting on the polynomials over Sn(V)⊕V⊕V*.

conjunctive See Sylvester (1853, Glossary p. 543–548)

connex A form in two sets of variables, one set corresponding to a vector space and the other to its dual, or in otherwords an element of the symmetric algebra of V⊕V* for a vector space V. Introduced by Clebsch.

continuant Main article: Continuant (mathematics)A determinant of a tridiagonal matrix.(Salmon 1885, p.18)

contragredient Transforming according to the dual of the natural representation of a linear group. (Elliott 1895,p.74)

contravariant A relative invariant of GL(V) acting on the polynomials over Sn(V)⊕V.

convolution A method of constructing invariants from two other invariants. (Glenn 1915, p.87)

covariancy (Elliott 1895, p.83)

covariant 1. (Noun) A relative invariant of GL(V) acting on the polynomials over Sn(V)⊕V*.

2. (Adjective) Invariant under the action of a group, especially for functions between two spaces acted on by thegroup.

cross ratio The cross ratio is an invariant of 4 points of a projective line.

cubic (Adjective) Degree 3

(Noun) A form of degree 3

cubicovariant A covariant of degree 3, in particular an order 3 degree 3 covariant of a binary cubic given by theJacobian of the cubic and its Hessian.. (Elliott 1895, p.50)

cubinvariant An invariant of degree 3.

cubo- Used to form compound adjectives such as cubo-linear, cubo-quadric, and so on, indicating the bidegree ofsomething. For example, cubo-linear means having degree 3 in the first of two sets of variables and degree 1in the second.

cumulant The numerator or denominator of a continued fraction, often expressed as a determinant. Sylvester (1853,Glossary p. 543–548).

10.6. D 93

10.6 Ddecic

decimic (Adjective) Degree 10

(Noun) A form of degree 10

degree 1. The degree of a form is the total power of the variables in it.

2. The degree of an invariant or covariant or contravariant means its degree in terms of the coefficients of the form.The degree of a form considered as a form is usually not its degree when considered as a covariant.

3. Some authors exchange the meanings of “degree” and “order” of a covariant or concomitant.

denary Depending on 10 variables

determinant The determinant is a joint invariant of n vectors of an n-dimensional space.

dialytic Sylvester’s dialytic method is a method for calculating resultants, essentially by expressing them as the de-terminant of a Sylvester matrix. See Sylvester (1853, Glossary p. 543–548). Archaic.

differentiant Another name for an invariant of a binary form. Archaic.

discriminant Main article: discriminantThe discriminant of a form in n variables is the multivariate resultant of the n differentials with respect to eachof the variables. For binary forms the discriminant vanishes if the form has multiple roots and is essentiallythe same as the discriminant of a polynomial of 1 variable. The discriminant of a form vanishes when thecorresponding hypersurface has singularities (as a scheme).

disjunctive See Sylvester (1853, Glossary p. 543–548)

divariant An alternative name for a concomitant suggested by Salmon (1885, p.121)

duodecic

duodecimic (Adjective) Degree 12

(Noun) A form of degree 12

10.7 Eeffective See Sylvester (1853, Glossary p. 543–548)

effluent See Sylvester (1853, Glossary p. 543–548). Archaic.

eliminant De Morgan’s name for the (multivariate) resultant, an invariant of n forms in n variables that vanishes ifthey have a common nonzero solution. (Elliott 1895, p.16)

emanant The rth emanant of a binary form in variables xi is a covariant given by the action of the rth power of thedifferential operator Σyi∂/∂xi. This is essentially the same as polarization. (Elliott 1895, p.56) Sylvester (1853,Glossary p. 543–548)

endoscopic See Sylvester (1853, Glossary p. 543–548). Archaic.

equianharmonic contravariant A weight 4 contravariant of binary quartics (Dolgachev 2012, 6.4)

evectant Main article: evectantA contravariant given by the action of an evector.

evector Main article: evectantA differential operator constructed from a binary form.

excess The excess of a polynomial in the coefficients a0,...ap of a form of degree p is ip–2w, where p is the degreeof the polynomial and w is its weight. (Elliott 1895, p.141)

94 CHAPTER 10. GLOSSARY OF INVARIANT THEORY

exoscopic See Sylvester (1853, Glossary p. 543–548). Archaic.

extensor An element of the kth exterior power of a vector space that can be written as the exterior product of kvectors.

extent The extent of a polynomial in a0, a1,... is the largest value of p such that the polynomial involves ap. (Elliott1895, p.138)

10.8 F

facient One of the variables of a form (Cayley 1860)

facultative A facultative point is one where a given function is positive. (Salmon 1885, p.243)

form A homogeneous polynomial in several variables, also called a quantic.

functional determinant An archaic name for Jacobians

fundamental 1. The first fundamental theorem describes generators (called brackets) for the ring of invariantpolynomials on a sum of copies of a vector space V and its dual (for the special linear group of V). The secondfundamental theorem describes the syzygies between the generators.

2. For fundamental scale see Sylvester (1853, Glossary p. 543–548). Archaic.

3. A fundamental invariant is an element of a set of generators for a ring of invariants.

4. A fundamental system is a set of generators (for a ring of invariants, covariants, and so on).

10.9 G

Gordan Named for Paul Gordan.

1. Gordan’s theorem states that the ring of invariants of a binary form (or several binary forms) is finitely generated.

grade The highest power of a bracket factor in the symbolic expression for an invariant. (Glenn 1915, 4.8)

gradient A homogeneous polynomial in a0, ..., ap all of whose terms have the same weight, where an has weight n.(Elliott 1895, p.138) Archaic.

Gröbner basis Main article: Gröbner basisA basis for an ideal of a ring of polynomials chosen according to some rule to make computations easier.

ground form An element of a minimal set of homogeneous generators for the invariants of a form. Archaic.

10.10 H

hectic A joke term for a form of degree 100.

harmonic contravariant A weight 6 contravariant of binary quartics (Dolgachev 2012, 6.4)

harmonizant A bilinear invariant of two forms whose vanishing means they are polar. (Dolgachev 2012, p.75)

Hermite Named after Charles Hermite

1. The Hermite contravariant is a degree 12 class 9 contravariant of ternary cubics. (Dolgachev 2012, 3.4.3)

2. Hermite’s law of reciprocity states that the degree m covariants of a binary form of degree n correspond to thedegree n covariants of a binary form of degree m.

3. The Hermite invariant is the degree 18 skew invariant of a binary quintic.

10.11. I 95

Hessian Main article: HessianA covariant of a form u, given by the determinant of the matrix with entries ∂2u/∂xi∂xj.

Hilbert Named after David Hilbert

A Hilbert series is a formal power series whose coefficients are dimensions of spaces of invariants of variousdegrees.

Hilbert’s theorem states that the ring of invariants of a finite-dimensional representation of a reductive group isfinitely generated.

homographic 1. A homographic transformation is a transformation taking x to (ax+b)/(cx+d).

2. A homographic relation between x and y is a relation of the form axy + bx + cy + d=0 .

hyperdeterminant Main article: hyperdeterminantAn invariant of a multidimensional array of coefficients, generalizing the determinant of a 2-dimensional array.

10.11 Iidentity covariant A form considered as a covariant of degree 1.

immanant Main article: immanant of a matrixA generalization of the determinant and permanent of a matrix

inertia The signature of a real quadratic form. See Sylvester (1853, Glossary p. 543–548)

integral rational function A polynomial.

intercalations See Sylvester (1853, Glossary p. 543–548). Archaic.

intermediate invariant An invariant of two forms constructed from two invariants of each of the forms. (Elliott1895, p.23)

intermutant A special form of permutant. (Cayley 1860)

invariant 1. (Adjective) Fixed by the action of a group

2. (Noun) An absolute invariant, meaning something fixed by a group action.

3. (Noun) A relative invariant, meaning something transforming according to a character of a group. In classicalinvariant theory it often refers to relatively invariant polynomials in the coefficients of a quantic, considered asa representation of a general linear group.

involutant See Sylvester’s collected papers, volume IV, page 135

irreducible Not expressible as a polynomial in things of smaller degree.

isobaric All terms having the same weight. (Elliott 1895, p.32)

10.12 JJacobian Main article: Jacobian matrix

A covariant of n forms fi in n variables xj, given by the determinant of the matrix with entries ∂fi/∂xj.

joint invariant A relative invariant for polynomials over reducible representation of a group, in particular a relativeinvariant for a several binary forms.

10.13 Kkenotheme Sylvester (1853, Glossary p. 543–548) defines this as “A finite system of discrete points defined by one

or more homogeneous equations in number one less than the number of variables contained therein.” This maymean an intersection of n hypersurfaces in n-dimensional projective space. Archaic.

96 CHAPTER 10. GLOSSARY OF INVARIANT THEORY

10.14 Llinear Degree 1

lineo- Used to form compound adjectives such as lineo-linear, lineo-quadric, and so on, indicating the bidegree ofsomething. For example, lineo-linear means having degree 1 in each of two sets of variables. In particular thelineo-linear invariant of two binary forms has degree 1 in the coefficients of each form. (Elliott 1895, p.54)

Lüroth invariant A degree 54 invariant vanishing on Lüroth quartics (nonsingular quartic plane curves containingthe 10 vertices of a complete pentalateral). (Dolgachev 2012, p.295)

10.15 Mmeicatalecticizant Sylvester’s original term for what he later renamed the catalecticant. Archaic.

mixed concomitant A concomitant that involves both covariant and contravariant variables, in other words one thatis not a covariant or contravariant. (Elliott 1895, p.77)

modular Main article: Modular invariant of a groupDefined over a finite field.

modulus An alternative name for the determinant of a linear transformation. (Elliott 1895, p.3)

monic 1. Adjective. Having leading coefficient 1.

2. Adjective. Having degree 1.

3. Noun. A form of degree 1.

monotheme See Sylvester (1853, Glossary p. 543–548). Archaic.

10.16 Nnonary Depending on 9 variables

nonic (Adjective) Degree 9

(Noun) A form of degree 9

nullcone The cone of nullforms

nullform Main article: nullformA form on which all invariants with zero constant term vanish.

10.17 Ooctavic

octic (Adjective) Degree 8

(Noun) A form of degree 8

octonary Depending on 8 variables

Omega process Main article: Cayley’s Ω process

order 1. The degree of a covariant or concomitant in the variables of a form.

2. Some authors interchange the meaning of “degree” and “order” of a covaraint.

10.18. P 97

3. See Sylvester (1853, Glossary p. 543–548)

ordinary An ordinary invariant means a relative invariant, in other words something transforming according to acharacter of a group, as opposed to an absolute invariant.

osculant Main article: osculantAn invariant of several forms of the same degree generalizing the tact-invariant of two forms, equal to thediscriminant if the number of forms is 1, and to the multivariate resultant if the number of forms is the numberof variables. Salmon (1885, p.171)

10.18 Ppartial transvectant Main article: transvectant

partition Main article: partition (number theory)An expression of a number as a sum of positive integers.(Elliott 1895, p.119)

peninvariant Same as seminvariant. (Cayley 1860)

permanent Main article: permanentA variation of the determinant of a matrix

permutant (Cayley 1860)

perpetuant Main article: perpetuantRoughly an irreducible covariant of a form of infinite order.

persymmetrical A persymmetrical matrix is a Hankel matrix. See Sylvester (1853, Glossary p. 543–548). Archaic.

Pfaffian Main article: PfaffianA square root of the determinant of a skew-symmetric matrix.

pippian An old name for the Cayleyan.

plagiogonal Related to or fixed by the orthogonal group of some quadratic form. See Sylvester’s collected papers,volume I, page 357

plexus A set of generators of an ideal, especially if the number of generators needed is larger than the codimensionof the corresponding variety.

polarization A method of reducing the degree of something by introducing extra variables.

principiant A reciprocant that is invariant under homographic substitutions, up to a constant facts. See Sylvester’scollected papers, vol IV, page 382

projective invariant 1. An invariant of the projective general linear group.

2. An invariant of a central extension of a group.

protomorph A set of protomorphs is a set of seminvariants, such that any seminvariant is a polynomial in theprotomorphs and the inverse of the first protomorph. (Elliott 1895, p.206)

10.19 Qquadratic

quadric (Adjective) Degree 2

(Noun) A form of degree 2

quadricovariant A covariant of degree 2. (Salmon 1885, p.261)

98 CHAPTER 10. GLOSSARY OF INVARIANT THEORY

quadrinvariant An invariant of degree 2. Sylvester (1853, Glossary p. 543–548).

quadro- Degree 2. Used to form compound adjectives such as quadro-linear, quadro-quadric, and so on, indicatingthe bidegree of something. For example, quadro-linear means having degree 2 in the first of two sets ofvariables and degree 1 in the other.

quantic An archaic name for a homogeneous polynomial in several variables, now usually called a form.

quartic (Adjective) Degree 4

(Noun) A form of degree 4

quarticovariant A covariant of degree 4.

quartinvariant An invariant of degree 4

quarto- Used to form compound adjectives such as quarto-linear, quarto-quadric, and so on, indicating the bidegreeof something. For example, quarto-linear means having degree 4 in the first of two sets of variables and degree1 in the other.

quaternary Depending on 4 variables

quinary Depending on 5 variables.

quintic (Adjective) Degree 5

(Noun) A form of degree 5

quintinvariant An invariant of degree 5.

quippian Main article: quippian

10.20 R

rational integral function A polynomial.

reciprocal The reciprocal of a matrix is the adjugate matrix.

reciprocant 1. A contravariant of a ternary form, giving the equation of a dual curve. (Elliott 1895, p.400)

reciprocity Exchanging the degree of a form with the degree of an invariant. For example, Hermite’s law of reci-procity states that the degree p invariants of a form of degree n correspond to the degree n invariants of a formof degree p. (Elliott 1895, p.137)

reducible Expressible as a polynomial in things of smaller degree.

relative invariant Something transforming according to a 1-dimensional character of a group, often a power of thedeterminant. Same as ordinary invariant.

resultant 1.Main article: resultantA joint invariant of two binary forms that vanishes when they have a common root. More generally a (multi-variate) resultant is a joint invariant of n forms in n variables that vanishes if they have a common nontrivialzero. Sometimes called an eliminant in older books.

2. An archaic term for the determinant

revenant Suggested by Sylvester (collected works vol 3, page 593) as an alternative name for a perpetuant.

Reynolds operator Main article: Reynolds operatorProjection onto the fixed vectors

rhizoristic See Sylvester (1853, Glossary p. 543–548). Archaic.

10.21. S 99

10.21 S

Salmon invariant A degree 60 invariant vanishing on ternary quartics with an inflection bitangent. (Dolgachev2012, 6.4)

Scorza covariant A covariant of ternary quartics. (Dolgachev 2012, 6.3.4)

semicovariant An analogue of seminvariants for covariants. See (Burnside & Panton 1881, p.329)

semi-invariant

seminvariant 1. The leading term of a covariant, also called its source. (Grace & Young 1903, section 33)

2. An invariant of the group of upper triangular matrices.

senary Depending on 6 variables. (Rare)

septenary Depending on 7 variables

septic

septimic (Adjective) Degree 7

(Noun) A form of degree 7

sextic (Adjective) Degree 6

(Noun) A form of degree 6

sexticovariant A covariant of degree 6

sextinvariant An invariant of degree 6 (Salmon 1885, p.262)

signaletic See Sylvester (1853, Glossary p. 543–548). Archaic.

singular 1. See Sylvester (1853, Glossary p. 543–548)

skew A skew invariant is a relative invariant of a group G that changes sign under an element of order 2 in itsabelianization. In particular for the general linear group it changes sign under elements of determinant –1, andfor the symmetric group it changes sign under odd permutations. For binary forms skew invariants are theinvariants of odd weight. They do not exist for binary quadrics, cubics, or quartics, but do for binary quintics.(Elliott 1895, p.112)

source The source of a covariant is its leading term, when the covariant is considered as a form. Also called aseminvariant. (Elliott 1895, p.126)

Steinerian Main article: Steinerian

symbolic Main article: symbolic methodThe symbolicmethod is a way of representing invariants, that repeatedly uses the identification of the symmetricpower of a vector space with the symmetric elements of a tensor power.

syrrhizoristic Sylvester (1853, Glossary p. 543–548) defined this as “A syrrhizoristic series is a series of discon-nected functions which serve to determine the effective intercalations of the real roots of two functions lyingbetween any assigned limits.” Archaic. This term does not seem to have been used (or understood) by anyoneother than Sylvester.

syzygant (Elliott 1895, p.198)

syzygetic See Sylvester (1853, Glossary p. 543–548)

syzygy A linear or algebraic relation, especially one between generators of a ring or module.

100 CHAPTER 10. GLOSSARY OF INVARIANT THEORY

10.22 Ttacinvariant

tact invariant An invariant of one or two ternary forms that vanishes if the corresponding curve touches itself, or ifthe two curves touch each other. It is generalized by the osculant.

tamisage Sylvester’s name for his method of guessing the degrees of a generating set of invariants or covariants byexamining the generating function.(Elliott 1895, p.175). Archaic.

tantipartite An archaic term for multilinear. (Cayley 1860)Tschirnhaus transformation Main article: Tschirnhaus transformation

ternary Depending on 3 variablesToeplitz invariant An invariant of nets of quadrics in 3-dimensional projective space that vanishes on nets with a

common polar pentahedron. (Dolgachev 2012, p.51)transfer A method of constructing contravariants of forms in n+1 variables from invariants of forms in n variables.

(Dolgachev 2012, 3.4.2)transvectant Main article: transvectant

An invariant formed from n invariants in n variables using Cayley’s omega process. (Elliott 1895, p.71)trinomial A polynomial with at most three non-zero coefficients.

10.23 Uueberschiebung Transvectant. (Elliott 1895, p.171)umbrae

umbral See Sylvester (1853, Glossary p. 543–548)unary Depending on 1 variable. Same as univariate.undecic

undecimic (Adjective) Degree 11(Noun) A form of degree 11unimodular Having determinant 1unitarian trick Finite-dimensional representations of a semisimple Lie group are equivalent to finite-dimensional

representations of a compact form, and are therefore completely reducible.univariate Depending on 1 variable. Same as unary.universal concomitant The pairing between a vector space and its dual, considered as a concomitant. (Elliott 1895,

p.77)

10.24 V

10.25 Wweight 1. The power of the determinant appearing in the formula for transformation of a relative invariant.2. A character of a torus3. See Sylvester (1853, Glossary p. 543–548)4. The weight of ai is i, and the weight of a product of monomials is the sum of their weights.

10.26. XYZ 101

10.26 XYZzeta

ζ A product of squared differences. See Sylvester (1853, Glossary p. 543–548)

10.27 See also• Glossary of classical algebraic geometry

10.28 References• Burnside, William Snow; Panton, Arthur William (1881), The theory of equations: With an introduction to thetheory of binary algebraic forms, 2 volumes, Hodges, Figgis & co., MR 0115987

• Dieudonné, Jean A.; Carrell, James B. (1970), “Invariant theory, old and new”, Advances in Mathematics 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525 Reprinted as Dieudonné, JeanA.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press, doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102

• Cayley, Arthur (1860), “Recent terminology in mathematics”, The English Cyclopaedia 5, Charles Knight,London, pp. 534–542, Reprinted in Cayley’s collected works, volume IV, pages 594–608

• Crilly, Tony (2006), Arthur Cayley. Mathematician laureate of the Victorian age, Johns Hopkins UniversityPress, ISBN 978-0-8018-8011-7, MR 2284396

• Dolgachev, Igor (2003), Lectures on invariant theory, London Mathematical Society Lecture Note Series 296,Cambridge University Press, doi:10.1017/CBO9780511615436, ISBN 978-0-521-52548-0, MR 2004511

• Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view (PDF), Cambridge University Press,ISBN 978-1-107-01765-8

• Elliott, Edwin Bailey (1895), An introduction to the algebra of quantics, Oxford, Clarendon Press, Reprintedby Chelsea Scientific Books 1964

• Glenn, Oliver E. (1915), A Treatise on the Theory of Invariants, Ginn and company, ISBN 978-1-4297-0030-6

• Grace, J. H.; Young, Alfred (1903), The algebra of invariants, Cambridge: Cambridge University Press

• Hilbert, David (1890), “Ueber die Theorie der algebraischen Formen”, Mathematische Annalen 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831

• Hilbert, D. (1893), "Über die vollen Invariantensysteme (On Full Invariant Systems)", Math. Annalen 42 (3):313, doi:10.1007/BF01444162

• Olver, Peter J. (1999), Classical invariant theory, Cambridge: Cambridge University Press, ISBN 0-521-55821-2

• Salmon, George (1885) [1859], Lessons introductory to the modern higher algebra (4th ed.), Dublin, Hodges,Figgis, and Co., ISBN 978-0-8284-0150-0

• Sylvester, James Joseph (1853), “On a Theory of the Syzygetic Relations of Two Rational Integral Functions,Comprising an Application to the Theory of Sturm’s Functions, and That of the Greatest Algebraical CommonMeasure”, Philosophical Transactions of the Royal Society of London (The Royal Society) 143: 407–548,doi:10.1098/rstl.1853.0018, ISSN 0080-4614, JSTOR 108572

• Sylvester, James Joseph; Franklin, F. (1879), “Tables of the Generating Functions and Groundforms for theBinary Quantics of the First Ten Orders”, American Journal of Mathematics (The Johns Hopkins UniversityPress) 2 (3): 223–251, doi:10.2307/2369240, ISSN 0002-9327

• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255

102 CHAPTER 10. GLOSSARY OF INVARIANT THEORY

10.29 External links• Brouwer, Andries E., Invariants of binary forms

Chapter 11

Glossary of Lie algebras

This is a glossary for the terminology applied in the mathematical theories of Lie algebras. The statements in thisglossary mainly focus on the algebraic sides of the concepts, without referring to Lie groups or other related subjects.

11.1 Definition

Lie algebra A vector space g over a field F with a binary operation [·, ·] (called the Lie bracket or abbr. bracket), which satisfies the following conditions: ∀a, b ∈ F, x, y, z ∈ g ,

1. [ax+ by, z] = a[x, z] + b[y, z] (bilinearity)2. [x, x] = 0 (alternating)3. [[x, y], z] + [[y, z], x] + [[z, x], y] = 0 (Jacobi identity)

associative algebra An associative algebraA can be made to a Lie algebra by defining the bracket [x, y] = xy−yx(the commutator of x, y ) ∀x, y ∈ A .

homomorphism A vector space homomorphism ϕ : g1 → g2 is said to be a Lie algebra homomorphism ifϕ([x, y]) = [ϕ(x), ϕ(y)]∀x, y ∈ g1.

adjoint representation Given x ∈ g , define map adx by

adx :g → g

y 7→ [x, y]

adx is a Lie algebra derivation. The map

ad :g → gl(g)

x 7→ adx

thus defined is a Lie algebra homomorphism.

ad : g → End(g) is called adjoint representation.

Jacobi identity The identity [[x, y], z] + [[y, z], x] + [[z, x], y] = 0.

To say Jacobi identity holds in a vector space is equivalent to say adjoint of all elements are derivations : adx([y, z]) =[adx(y), z] + [y, adx(z)] .

103

104 CHAPTER 11. GLOSSARY OF LIE ALGEBRAS

11.1.1 subalgebras

subalgebra A subspace g′ of a Lie algebra g is called the subalgebra of g if it is closed under bracket, i.e. [g′, g′] ⊆ g′.

ideal A subspace g′ of a Lie algebra g is the ideal of g if [g′, g] ⊆ g′.

In particular, every ideal is also a subalgebra. Every kernel of a Lie algebra homomorphism is an ideal. Unlike inring theory, there is no distinguishability of left ideal and right ideal.

derived algebra The derived algebra of a Lie algebra g is [g, g] . It is a subalgebra.

normalizer The normalizer of a subspaceK of a Lie algebra g is Ng(K) := {x ∈ g|[x,K] ⊆ K} .

centralizer The centralizer of a subset X of a Lie algebra g is Cg(X) := {x ∈ g|[x,X] = {0}} .

center The center of a Lie algebra is the centralizer of itself : Z(L) := {x ∈ g|[x, g] = 0}

radical The radical Rad(g) is the maximum solvable ideal of g .

11.2 Solvability, nilpotency, Jordan decomposition, semisimplicityabelian A Lie algebra is said to be abelian if its derived algebra is zero.

nilpotent Lie algebra A Lie algebra L is said to be nilpotent if CN (L) = {0} for some positive integer N .

The following conditions are equivalent:

• CN (L) = {0} for some positive integer N , i.e. the descending central series eventually terminates to{0} .

• CN (L) = L for some positive integer N, i.e. the ascending central series eventually terminates to L.• There exists a chain of ideals of L , L = I1 ⊇ I2 ⊇ I3 ⊇ · · · ⊇ In = {0} , such that [L, Ik] ⊆ Ik+1 .• There exists chain of ideals ofL , L = I1 ⊇ I2 ⊇ I3 · · · ⊇ In = {0} , such that Ik/Ik+1 ⊆ Z(L/Ik+1).

• adx is nilpotent ∀x ∈ L . (Engel’s theorem)• adL is a nilpotent Lie algebra.

In particular, every nilpotent Lie algebra is solvable.

If L is nilpotent, any subalgebra and quotient of L are nilpotent.

nilpotent element in a Lie algebra An element x ∈ L is said to be nilpotent in L if adx is a nilpotentendomorphism, i.e. viewing adx as a matrix in glg , ∃N ∈ Z+, (adx)N = 0 . It is equivalent to(adx)Ny = [x[x . . . [x[x, y] . . .] = 0 ∀y ∈ L

descending central series a sequence of ideals of a Lie algebra L defined by C0(L) = L, C1(L) =[L,L], Cn+1(L) = [L,Cn(L)]

ascending central series a sequence of ideals of a Lie algebra L defined by C0(L) = L, C1(L) =Z(L) (center of L) , Cn+1(L) = π−1

n (Z(L/Cn(L))) , where πi is the natural homomorphismL → L/Cn(L)

solvable Lie algebra ALie algebraL is said to be solvable ifL(N) = 0 for some positive integerN , i.e. the derivedseries eventually terminates to {0} .

The following condition is equivalent to solvability:

11.3. SEMISIMPLE LIE ALGEBRA 105

* There exists chain of ideals of L , L = I1 ⊇ I2 ⊇ I3 · · · ⊇ In = {0} , such that [Ik, Ik] ⊆ Ik+1 .

If L is solvable, any subalgebra and quotient of L are solvable.Let I is an ideal of a Lie algebra L . If L/I, I are solvable, L is solvable.

derived series a sequence of ideals of a Lie algebra L defined by L(0) = L, L(1) = [L,L], L(n+1) =[L(n), L(n)]

simple A Lie algebra is said to be simple if it is non-abelian and has only two ideals, itself and {0} .

semisimple Lie algebra A Lie algebra is said to be semisimple if its radical is {0} .

semisimple element in a Lie algebra

split Lie algebra

free Lie algebra

toral Lie algebra

Lie’s theorem Let g be a finite-dimensional complex solvable Lie algebra over algebraically closed field of charac-teristic 0 , and let V be a nonzero finite dimensional representation of g . Then there exists an element of Vwhich is a simultaneous eigenvector for all elements of g .

Corollary: There exists a basis of V with respect to which all elements of g are upper triangular.

Killing form The Killing form on a Lie algebra g is a symmetric, associative, bilinear form defined by κ(x, y) :=Tr(adx ad y) ∀x, y ∈ g .

Cartan criterion for solvability A Lie algebra g is solvable iff κ(g, [g, g]) = 0 .

Cartan criterion for semisimplity If κ(·, ·) is nondegenerate, then g is semisimple.

If g is semisimple and the underlying field F has characteristic 0 , then κ(·, ·) is nondegenerate.

lower central series synonymous to “descending central series”.

upper central series synonymous to “ascending central series”.

11.3 Semisimple Lie algebra

11.4 Root System (for classification of semisimple Lie algebra)In the below section, denote (·, ·) as the inner product of a Euclidean space E.In the below section, < ·, · > denoted the function defined as < β,α >= (β,α)

(α,α) ∀α, β ∈ E .

Cartan subalgebra A Cartan subalgebra h of a Lie algebra g is a nilpotent subalgebra satisfying Ng(h) = h .

regular element of a Lie algebra

maximal toral subalgebra

Borel subalgebra

106 CHAPTER 11. GLOSSARY OF LIE ALGEBRAS

root of a semisimple Lie algebra Let g be a semisimple Lie algebra, h be a Cartan subalgebra of g . For α ∈ h∗ ,let gα := {x ∈ g|[h, x] = α(h)x ∀h ∈ h} . \alpha is called a root of g if it is nonzero and gα = {0}

The set of all roots is denoted by Φ ; it forms a root system.

Root system A subset Φ of the Euclidean space E is called a root system if it satisfies the following conditions:

• Φ is finite, span(Φ) = E and 0 /∈ Φ .• For all α ∈ Φ and c ∈ R , cα ∈ Φ iff c = ±1 .• For all α, β ∈ Φ , < α, β > is an integer.• For all α, β ∈ Φ , Sα(β) ∈ Φ , where Sα is reflection through hyperplane normal to α i.e. Sα(x) =x− < x,α > α .

Cartan matrix Cartan matrix of root system Φ is matrix (< αi, αj >)ni,j=1 where ∆ = {α1 . . . αn} is a set ofsimple roots of Φ .

Dynkin diagrams

Simple Roots A subset∆ of a root system Φ is called a set of simple roots if it satisfies the following conditions:

• ∆ is linear basis of E .• Each element of Φ is a linear combination of elements of ∆ with coefficients which are either all non-negative or all nonpositive.

a partial order on the Eucliean space E defined by the set of simple root∀λ, µ ∈ E, λ > µ ⇐⇒ λ− µ > 0 ⇐⇒ ∃k1, k2, ..., kn ∈ Z+, α1, α2, ..., αn ∈ ∆, λ− µ =

∑i kiαi

regular element with respect to a root system LetΦ be a root system. γ ∈ E is called regular if (γ, α) = 0∀γ ∈Φ .

For each set of simple roots ∆ of Φ , there exists a regular element γ ∈ E such that (γ, α) > 0∀γ ∈ ∆ ,conversely for each regular γ there exist a unique set of base roots∆(γ) such that the previous condition holdsfor∆ = ∆(γ) . It can be determined in following way: letΦ+(γ) = {α ∈ Φ|(α, γ) > 0} . Call an element αof Φ+(γ) decomposable if α = α′ + α′′ where α′, α′′ ∈ Φ+(γ) , then∆(γ) is the set of all indecomposableelements of Φ+(γ)

positive roots Positive root of root system Φ with respect to a set of simple roots∆ is a root of Φ which is a linearcombination of elements of∆ with nonnegative coefficients.

negative roots Negative root of root system Φ with respect to a set of simple roots∆ is a root of Φ which is a linearcombination of elements of∆ with nonpositive coefficients.

long root

short root

Weyl group Weyl group of a root system Φ is a (necessarily finite) group of orthogonal linear transformations of Ewhich is generated by reflections through hyperplanes normal to roots of Φ

inverse of a root system Given a root system Φ . Define αv = 2α(α,α) , Φv = {αv|α ∈ Φ} is called the inverse of a

root system.

Φv is again a root system and have the identical Weyl group as Φ .

base of a root system synonymous to “set of simple roots”

dual of a root system synonymous to “inverse of a root system”

11.5. REPRESENTATION THEORY 107

11.4.1 theory of weights

<--

weight in a root system λ ∈ E is called a weight if < λ,α >∈ Z ∀α ∈ Φ . -->

weight lattice

weight space

dominant weight A weight \lambda is dominant if < λ,α >∈ Z+ for some α ∈ Φ

fundamental dominant weight Given a set of simple roots∆ = {α1, α2, ..., αn} , it is a basis ofE . αv1, α

v2, ..., α

vn ∈

Φv is a basis of E too; the dual basis λ1, λ2, ..., λn defined by (λi, αvj ) = δij , is called the fundamental dom-

inant weights.

highest weight

minimal weight

multiplicity (of weight)

radical weight

strongly dominant weight

11.5 Representation theorymodule Define an action of g on a vector space V ( i.e. an operation g × V → V, (x, v) 7→ xv ) such that:

∀a, b ∈ F, x, y ∈ g, v, w ∈ V satisfy

# (ax+ by)v = a(xv) + b(yv)

# x(av + bw) = a(xv) + b(xw)

# [x, y]v = x(yv)− y(xv)

Then V is called a g -module. (Remark: V, g have the same underlying field F .)

Each g -module corresponds to a representation g → glV .

A subspace W is a submodule (more precisely, sub g -module) of V if g -moduleW ⊂ V .

representation For a vector space V , if there is a Lie algebra homomorphism π : g → glV , then π is called arepresentation of g .

Each representation g → glV corresponds to a g -module V .

A subrepresentation is the representation corresponding to a submodule.

homomorphism Given two g -module V, W, a g -module homomorphism ϕ is a vector space homomorphismsatisfying ϕ(xv) = xϕ(v)∀x ∈ g, v ∈ V .

trivial representation A representation is said to be trivial if the image of g is the zero vector space. It correspondsto the action of g on module V by xv = 0∀x ∈ g, v ∈ V .

faithful representation If the representation g → glV is injective, it is said to be faithful.

108 CHAPTER 11. GLOSSARY OF LIE ALGEBRAS

tautology representation If a Lie algebra g is defined as a subalgebra of gl(n, F ) , like sl(n, F ), o(2l, F ), t(n, F )(the upper triangular matrices), the tautology representation is the imbedding g → gl(n, F ) . It correspondsto the action on module Fn by the matrix multipation.

adjoint representation The representationad :g → glg

x 7→ adx. It corresponds viewing g as a g -module - the action on

the module is given by the adjoint endomorphism.

irreducible modules A module is said to be irreducible if it has only two submodules, itself and zero.

indecomposable module A module is said to be indecomposable if it cannot be written as direct sum of two non-zero submodules.

An irreducible module need not be indecmoposable but the converse is not true.

completely reducible module A module is said to be completely reducible if it can be written as direct sum ofirreducible modules.

simple module Synonymous as irreduible module.

quotient module / quotient representation Given a g -module V and its submodule W, an action g on V/W can bedefined by x(v +W ) = xv +W ∀x ∈ g, v ∈ V . V/W is said to be a quotient module in this case.

Schur’s lemma Statement in the language of module theory: Given V an irreducible g -module, ϕV → V is a g-module homomorphism iff ϕ = λ1V for some λ ∈ F .

Statement in the language of representation theory: Given an irreducible representation ϕ : L → gl(V ) , for θ ∈End(V ) , θϕ(x) = ϕ(x)θ iff θ = λ1V for some λ ∈ F .

simple module synonymous to “irreduible module”.

factor module synonymous to “quotient module”.

11.5.1 Universal enveloping algebras

PBW theorem (Poincaré–Birkhoff–Witt theorem)

11.5.2 Verma modules

BGG category \mathcal{O}

11.6 cohomology

11.7 Chevalley basis

a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley usedthese bases to construct analogues of Lie groups over finite fields, called Chevalley groups.The generators of a Lie group are split into the generators H and E such that:

[Hαi ,Hαj ] = 0

[Hαi , Eαj ] = AijEαj

[Eαi , Eαj ] = Hαj

[Eβ , Eγ ] = ±(p+ 1)Eβ+γ

where p = m if β + γ is a root and m is the greatest positive integer such that γ − mβ is a root.

11.8. EXAMPLES OF LIE ALGEBRA 109

11.8 Examples of Lie algebrageneral linear algebra gl(n, F )

Ado’s theorem Any finite-dimensional Lie algebra is isomorphic to a subalgebra of glV for some finite-dimensionalvector space V.

11.8.1 complex Lie algebras of 1D, 2D, 3D

11.8.2 Simple Algebras

Classical Lie algebras:Exceptional Lie algebras:

11.8.3 Miscellaneous

• Poisson algebra

• Kac–Moody algebra

11.9 Other discipline related• Lie group

• Glossary of semisimple groups

• Linear algebraic group

• Particle physics and representation theory

11.10 References• Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0

• Humphreys, James E. Introduction to Lie Algebras and Representation Theory, Second printing, revised. Grad-uate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. ISBN 0-387-90053-5

Chapter 12

Glossary of module theory

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of thesubject.

12.1 Basic definitionleft R-module A left moduleM over the ringR is an abelian group (M,+)with an operationR×M → M (called

scalar multipliction) satisfies the following condition:

∀r, s ∈ R,m, n ∈ M

1. r(m+ n) = rm+ rn

2. r(sm) = (rs)m

3. 1R m = m

right R-module A right module M over the ring R is an abelian group (M,+) with an operation M × R → Msatisfies the following condition:

∀r, s ∈ R,m, n ∈ M

1. (m+ n)r = mr + nr

2. (ms)r = r(sm)

3. m1R = m

Or it can be defined as the left moduleM over Rop (the opposite ring of R ).

bimodule If an abelian groupM is both a left S -module and rightR -module, it can be made to a (S,R) -bimoduleif s(mr) = (sm)r ∀s ∈ S, r ∈ R,m ∈ M .

submodule GivenM is a left R -module, a subgroup N ofM is a submodule if RN ⊆ N .

homomorphism of R -modulesFor two left R -modules M1,M2 , a group homomorphism ϕ : M1 → M2 is called homomorphism of R -modules if rϕ(m) = ϕ(rm)∀r ∈ R,m ∈ M1 .

quotient module Given a left R -modulesM , a submoduleN ,M/N can be made to a left R -module by r(m+N) = rm+N ∀r ∈ R,m ∈ M . It is also called a factor module.

annihilator The annihilator of a left R -module M is the set Ann(M) := {r ∈ R|rm = 0 ∀m ∈ M} . It is a(left) ideal of R .

The annihilator of an elementm ∈ M is the set Ann(m) := {r ∈ R|rm = 0} .

110

12.2. TYPES OF MODULES 111

12.2 Types of modulesfinitely generated module A module M is finitely generated if there exist finitely many elements x1, ..., xn in M

such that every element ofM is a finite linear combination of those elements with coefficients from the scalarring R .

cyclic module A module is called a cyclic module if it is generated by one element.

free module A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum ofcopies of the scalar ring R .

basis A basis of a moduleM is a set of elements inM such that every element in the module can be expressedas a finite sum of elements in the basis in a unique way.

Projective module AR -module P is called a projective module if given aR -module homomorphism g : P → M, and a surjectiveR -module homomorphism f : N → M , there exists aR -module homomorphism h : P →N such that f ◦ h = g .

The characteristic property of projective modules is called lifting.

The following conditions are equivalent:

• The covariant functor HomR(P,−) is exact.

112 CHAPTER 12. GLOSSARY OF MODULE THEORY

• M is a projective module.• Every short exact sequence 0 → L → L′ → P → 0 is split.• M is a direct summand of free modules.

In particular, every free module is projective.

injective module A R -module Q is called an injective module if given a R -module homomorphism g : X → Q ,and an injective R -module homomorphism f : X → Y , there exists a

R -module homomorphism h : Y → Q such that f ◦ h = g .

The module Q is injective if the diagram commutes

The following conditions are equivalent:

• The contravariant functor HomR(−, I) is exact.• I is a injective module.• Every short exact sequence 0 → I → L → L′ → 0 is split.

flat module A R -module F is called a flat module if the tensor product functor −⊗R F is exact.

In particular, every projective module is flat.

simple module A simple module is a nonzero module whose only submodules are zero and itself.

indecomposable module An indecomposable module is a non-zero module that cannot be written as a direct sumof two non-zero submodules. Every simple module is indecomposable.

12.3. OPERATIONS ON MODULES 113

principal indecomposable module A cyclic indecomposable projective module is known as a PIM.

semisimple module A module is called semisimple if it is the direct sum of simple submodules.

faithful module A faithful moduleM is one where the action of each nonzero r ∈ R onM is nontrivial (i.e. rx = 0for some x in M). Equivalently, Ann(M) is the zero ideal.

Noetherian module ANoetherian module is a module such that every submodule is finitely generated. Equivalently,every increasing chain of submodules becomes stationary after finitely many steps.

Artinian module An Artinian module is a module in which every decreasing chain of submodules becomes station-ary after finitely many steps.

finite length module A module which is both Artinian and Noetherian has additional special properties.

graded module A module M over a graded ring A =⊕

n∈N An is a graded module if M can be expressed as adirect sum

⊕i∈N Mi and AiMj ⊆ Mi+j .

invertible module Roughly synonymous to rank 1 projective module.

uniform module Module in which every two non-zero submodules have a non-zero intersection.

algebraically compact module (pure injective module) Modules in which all systems of equations can be decidedby finitary means. Alternatively, those modules which leave pure-exact sequence exact after applying Hom.

injective cogenerator An injective module such that every module has a nonzero homomorphism into it.

irreducible module synonymous to “simple module”

completely reducible module synonymous to “semisimple module”

12.3 Operations on modulesDirect sum of modules

Tensor product of modules

Hom functor

Ext functor

Tor functor

Essential extension An extension in which every nonzero submodule of the larger module meets the smaller modulein a nonzero submodule.

Injective envelope A maximal essential extension, or a minimal embedding in an injective module

Projective cover A minimal surjection from a projective module.

Socle The largest semisimple submodule

Radical of a module The intersection of the maximal submodules. For Artinian modules, the smallest submodulewith semisimple quotient.

114 CHAPTER 12. GLOSSARY OF MODULE THEORY

12.3.1 Changing scalars

Restriction of scalars Uses a ring homomorphism from R to S to convert S-modules to R-modules

Extension of scalars Uses a ring homomorphism from R to S to convert R-modules to S-modules

Localization of a module Converts R modules to S modules, where S is a localization of R

Endomorphism ring A left R-module is a right S-module where S is its endomorphism ring.

12.4 Homological algebra

Mittag-Leffler condition (ML)

Short five lemma

Five lemma

Snake lemma

12.5 Modules over special rings

D-module A module over a ring of differential operators.

Drinfeld module A module over a ring of functions on algebraic curve with coefficients from a finite field.

Galois module A module over the group ring of a Galois group

Structure theorem for finitely generated modules over a principal ideal domainFinitely generated modules over PIDs are finite direct sums of primary cyclic modules.

Tate module A special kind of Galois module

12.6 Miscellaneous

Rational canonical form

elementary divisor

invariants

fitting ideal

normal forms for matrices

Jordan Hölder composition series

tensor product

12.7 See also

• Glossary of ring theory

12.8. REFERENCES 115

12.8 References• John A. Beachy (1999). Introductory Lectures on Rings and Modules (1st ed.). Addison-Wesley. ISBN 0-521-64407-0.

• Golan, Jonathan S.; Head, Tom (1991),Modules and the structure of rings, Monographs and Textbooks in Pureand Applied Mathematics 147, Marcel Dekker, ISBN 978-0-8247-8555-0, MR 1201818

• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

• Serge Lang (1993). Algebra (3rd ed.). Addison-Wesley. ISBN 0-201-55540-9.

• Passman, Donald S. (1991), A course in ring theory, The Wadsworth & Brooks/Cole Mathematics Series,Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, ISBN 978-0-534-13776-2, MR1096302

Chapter 13

Glossary of order theory

This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice,and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources mightbe the following overview articles:

• completeness properties of partial orders

• distributivity laws of order theory

• preservation properties of functions between posets.

In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning isclear from the context, ≤ will suffice to denote the corresponding relational symbol, even without prior introduction.Furthermore, < will denote the strict order induced by ≤.Contents :

• Top

• 0–9

• A

• B

• C

• D

• E

• F

• G

• H

• I

• J

• K

• L

• M

• N

116

13.1. A 117

• O

• P

• Q

• R

• S

• T

• U

• V

• W

• X

• Y

• Z

13.1 A• Acyclic. A binary relation is acyclic if it contains no “cycles": equivalently, its transitive closure is antisymmetric.[1]

• Adjoint. See Galois connection.

• Alexandrov topology. For a preordered set P, any upper set O is Alexandrov-open. Inversely, a topology isAlexandrov if any intersection of open sets is open.

• Algebraic poset. A poset is algebraic if it has a base of compact elements.

• Antichain. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinctelements x and y such that x ≤ y. In other words, the order relation of an antichain is just the identity relation.

• Approximates relation. See way-below relation.

• A relation R on a set X is antisymmetric, if x R y and y R x implies x = y, for all elements x, y in X.

• An antitone function f between posets P and Q is a function for which, for all elements x, y of P, x ≤ y (inP) implies f(y) ≤ f(x) (in Q). Another name for this property is order-reversing. In analysis, in the presenceof total orders, such functions are often called monotonically decreasing, but this is not a very convenientdescription when dealing with non-total orders. The dual notion is called monotone or order-preserving.

• Asymmetric. A relation R on a set X is asymmetric, if x R y implies not y R x, for all elements x, y in X.

• An atom in a poset P with least element 0, is an element that is minimal among all elements that are unequalto 0.

• A atomic poset P with least element 0 is one in which, for every non-zero element x of P, there is an atom aof P with a ≤ x.

13.2 B• Base. See continuous poset.

• A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every elementx has a complement ¬x, such that x ∧ ¬x = 0 and x ∨ ¬x = 1.

• A bounded poset is one that has a least element and a greatest element.

• A poset is bounded complete if every of its subsets with some upper bound also has a least such upper bound.The dual notion is not common.

118 CHAPTER 13. GLOSSARY OF ORDER THEORY

13.3 C• Chain. A chain is a totally ordered set or a totally ordered subset of a poset. See also total order.

• Chain complete. A partially ordered set in which every chain has a least upper bound.

• Closure operator. A closure operator on the poset P is a function C : P → P that is monotone, idempotent,and satisfies C(x) ≥ x for all x in P.

• Compact. An element x of a poset is compact if it is way below itself, i.e. x<<x. One also says that such an xis finite.

• Comparable. Two elements x and y of a poset P are comparable if either x ≤ y or y ≤ x.

• Comparability graph. The comparability graph of a poset (P, ≤) is the graph with vertex set P in whichthe edges are those pairs of distinct elements of P that are comparable under ≤ (and, in particular, under itsreflexive reduction <).

• Complete Boolean algebra. A Boolean algebra that is a complete lattice.

• Complete Heyting algebra. A Heyting algebra that is a complete lattice is called a complete Heyting algebra.This notion coincides with the concepts frame and locale.

• Complete lattice. A complete lattice is a poset in which arbitrary (possibly infinite) joins (suprema) and meets(infima) exist.

• Complete partial order. A complete partial order, or cpo, is a directed complete partial order (q.v.) withleast element.

• Complete relation. Synonym for Total relation.

• Complete semilattice. The notion of a complete semilattice is defined in different ways. As explained in thearticle on completeness (order theory), any poset for which either all suprema or all infima exist is already acomplete lattice. Hence the notion of a complete semilattice is sometimes used to coincide with the one of acomplete lattice. In other cases, complete (meet-) semilattices are defined to be bounded complete cpos, whichis arguably the most complete class of posets that are not already complete lattices.

• Completely distributive lattice. A complete lattice is completely distributive if arbitrary joins distribute overarbitrary meets.

• Completion. A completion of a poset is an order-embedding of the poset in a complete lattice.

• Continuous poset. A poset is continuous if it has a base, i.e. a subset B of P such that every element x of Pis the supremum of a directed set contained in {y in B | y<<x}.

• Continuous function. See Scott-continuous.

• Converse. The converse <° of an order < is that in which x <° y whenever y < x.

• Cover. An element y of a poset P is said to cover an element x of P (and is called a cover of x) if x < y andthere is no element z of P such that x < z < y.

• cpo. See complete partial order.

13.4 D• dcpo. See directed complete partial order.

• A dense poset P is one in which, for all elements x and y in P with x < y, there is an element z in P, such thatx < z < y. A subset Q of P is dense in P if for any elements x < y in P, there is an element z in Q such that x <z < y.

• Directed. A non-empty subset X of a poset P is called directed, if, for all elements x and y of X, there is anelement z of X such that x ≤ z and y ≤ z. The dual notion is called filtered.

13.5. E 119

• Directed complete partial order. A posetD is said to be a directed complete poset, or dcpo, if every directedsubset of D has a supremum.

• Distributive. A lattice L is called distributive if, for all x, y, and z in L, we find that x ∧ (y ∨ z) = (x ∧ y) ∨(x ∧ z). This condition is known to be equivalent to its order dual. A meet-semilattice is distributive if for allelements a, b and x, a ∧ b ≤ x implies the existence of elements a' ≥ a and b' ≥ b such that a' ∧ b' = x. Seealso completely distributive.

• Domain. Domain is a general term for objects like those that are studied in domain theory. If used, it requiresfurther definition.

• Down-set. See lower set.

• Dual. For a poset (P, ≤), the dual order Pd = (P, ≥) is defined by setting x ≥ y if and only if y ≤ x. Thedual order of P is sometimes denoted by Pop, and is also called opposite or converse order. Any order theoreticnotion induces a dual notion, defined by applying the original statement to the order dual of a given set. Thisexchanges ≤ and ≥, meets and joins, zero and unit.

13.5 E

• Extension. For partial orders ≤ and ≤′ on a set X, ≤′ is an extension of ≤ provided that for all elements x andy of X, x ≤ y implies that x ≤′ y.

13.6 F

• Filter. A subset X of a poset P is called a filter if it is a filtered upper set. The dual notion is called ideal.

• Filtered. A non-empty subset X of a poset P is called filtered, if, for all elements x and y of X, there is anelement z of X such that z ≤ x and z ≤ y. The dual notion is called directed.

• Finite element. See compact.

• Frame. A frame F is a complete lattice, in which, for every x in F and every subset Y of F, the infinitedistributive law x ∧

∨Y =

∨{x ∧ y | y in Y} holds. Frames are also known as locales and as complete Heyting

algebras.

13.7 G

• Galois connection. Given two posets P and Q, a pair of monotone functions F:P→ Q and G:Q→ P is called aGalois connection, if F(x) ≤ y is equivalent to x ≤ G(y), for all x in P and y in Q. F is called the lower adjointof G and G is called the upper adjoint of F.

• Greatest element. For a subset X of a poset P, an element a of X is called the greatest element of X, if x ≤ afor every element x in X. The dual notion is called least element.

• Ground set. The ground set of a poset (X, ≤) is the set X on which the partial order ≤ is defined.

13.8 H

• Heyting algebra. A Heyting algebra H is a bounded lattice in which the function fa: H → H, given by fa(x)= a ∧ x is the lower adjoint of a Galois connection, for every element a of H. The upper adjoint of fa is thendenoted by ga, with ga(x) = a⇒ x. Every Boolean algebra is a Heyting algebra.

• Hasse diagram. A Hasse diagram is a type of mathematical diagram used to represent a finite partially orderedset, in the form of a drawing of its transitive reduction.

120 CHAPTER 13. GLOSSARY OF ORDER THEORY

13.9 I

• An ideal is a subset X of a poset P that is a directed lower set. The dual notion is called filter.

• The incidence algebra of a poset is the associative algebra of all scalar-valued functions on intervals, withaddition and scalar multiplication defined pointwise, and multiplication defined as a certain convolution; seeincidence algebra for the details.

• Infimum. For a poset P and a subset X of P, the greatest element in the set of lower bounds of X (if it exists,which it may not) is called the infimum,meet, or greatest lower bound of X. It is denoted by inf X or

∧X.

The infimum of two elements may be written as inf{x,y} or x ∧ y. If the set X is finite, one speaks of a finiteinfimum. The dual notion is called supremum.

• Interval. For two elements a, b of a partially ordered set P, the interval [a,b] is the subset {x in P | a ≤ x ≤ b}of P. If a ≤ b does not hold the interval will be empty.

• Interval finite poset. A partially ordered set P is interval finite if every interval of the form {x in P | x ≤ a}is a finite set.[2]

• Inverse. See converse.

• Irreflexive. A relation R on a set X is irreflexive, if there is no element x in X such that x R x.

• Isotone. See monotone.

13.10 J

• Join. See supremum.

13.11 L

• Lattice. A lattice is a poset in which all non-empty finite joins (suprema) and meets (infima) exist.

• Least element. For a subset X of a poset P, an element a of X is called the least element of X, if a ≤ x forevery element x in X. The dual notion is called greatest element.

• The length of a chain is the number of elements less one. A chain with 1 element has length 0, one with 2elements has length 1, etc.

• Linear. See total order.

• Linear extension. A linear extension of a partial order is an extension that is a linear order, or total order.

• Locale. A locale is a complete Heyting algebra. Locales are also called frames and appear in Stone duality andpointless topology.

• Locally finite poset. A partially ordered set P is locally finite if every interval [a, b] = {x in P | a ≤ x ≤ b} isa finite set.

• Lower bound. A lower bound of a subset X of a poset P is an element b of P, such that b ≤ x, for all x in X.The dual notion is called upper bound.

• Lower set. A subset X of a poset P is called a lower set if, for all elements x in X and p in P, p ≤ x impliesthat p is contained in X. The dual notion is called upper set.

13.12. M 121

13.12 M• Maximal chain. A chain in a poset to which no element can be added without losing the property of beingtotally ordered. This is stronger than being a saturated chain, as it also excludes the existence of elements eitherless than all elements of the chain or greater than all its elements. A finite saturated chain is maximal if andonly if it contains both a minimal and a maximal element of the poset.

• Maximal element. A maximal element of a subset X of a poset P is an element m of X, such that m ≤ ximplies m = x, for all x in X. The dual notion is called minimal element.

• Meet. See infimum.

• Minimal element. A minimal element of a subset X of a poset P is an elementm of X, such that x ≤m impliesm = x, for all x in X. The dual notion is called maximal element.

• Monotone. A function f between posets P and Q is monotone if, for all elements x, y of P, x ≤ y (in P) impliesf(x) ≤ f(y) (in Q). Other names for this property are isotone and order-preserving. In analysis, in the presenceof total orders, such functions are often called monotonically increasing, but this is not a very convenientdescription when dealing with non-total orders. The dual notion is called antitone or order reversing.

13.13 O• Order-dual. The order dual of a partially ordered set is the same set with the partial order relation replacedby its converse.

• Order-embedding. A function f between posets P and Q is an order-embedding if, for all elements x, y of P,x ≤ y (in P) is equivalent to f(x) ≤ f(y) (in Q).

• Order isomorphism. A mapping f: P → Q between two posets P and Q is called an order isomorphism,if it is bijective and both f and f−1 are monotone. Equivalently, an order isomorphism is a surjective orderembedding.

• Order-preserving. See monotone.

• Order-reversing. See antitone.

13.14 P• Partial order. A partial order is a binary relation that is reflexive, antisymmetric, and transitive. In a slightabuse of terminology, the term is sometimes also used to refer not to such a relation, but to its correspondingpartially ordered set.

• Partially ordered set. A partially ordered set (P, ≤), or poset for short, is a set P together with a partial order ≤on P.

• Poset. A partially ordered set.

• Preorder. A preorder is a binary relation that is reflexive and transitive. Such orders may also be calledquasiorders. The term preorder is also used to denote an acyclic binary relation (also called an acyclic digraph).

• Preserving. A function f between posets P and Q is said to preserve suprema (joins), if, for all subsets X of Pthat have a supremum sup X in P, we find that sup{f(x): x in X} exists and is equal to f(sup X). Such a functionis also called join-preserving. Analogously, one says that f preserves finite, non-empty, directed, or arbitraryjoins (or meets). The converse property is called join-reflecting.

• Prime. An ideal I in a lattice L is said to be prime, if, for all elements x and y in L, x ∧ y in I implies x in I ory in I. The dual notion is called a prime filter. Equivalently, a set is a prime filter if and only if its complementis a prime ideal.

• Principal. A filter is called principal filter if it has a least element. Dually, a principal ideal is an ideal with agreatest element. The least or greatest elements may also be called principal elements in these situations.

122 CHAPTER 13. GLOSSARY OF ORDER THEORY

• Projection (operator). A self-map on a partially ordered set that is monotone and idempotent under functioncomposition. Projections play an important role in domain theory.

• Pseudo-complement. In a Heyting algebra, the element x ⇒ 0 is called the pseudo-complement of x. It isalso given by sup{y : y ∧ x = 0}, i.e. as the least upper bound of all elements y with y ∧ x = 0.

13.15 Q• Quasiorder. See preorder.

• Quasitransitive. A relation is quasitransitive if the relation on distinct elements is transitive. Transitive impliesquasitransitive and quasitransitive implies acyclic.[1]

13.16 R• Reflecting. A function f between posets P and Q is said to reflect suprema (joins), if, for all subsets X of Pfor which the supremum sup{f(x): x in X} exists and is of the form f(s) for some s in P, then we find that supX exists and that sup X = s . Analogously, one says that f reflects finite, non-empty, directed, or arbitrary joins(or meets). The converse property is called join-preserving.

• Reflexive. A binary relation R on a set X is reflexive, if x R x holds for all elements x, y in X.

• Residual. A dual map attached to a residuated mapping.

• Residuated mapping. A monotone map for which the preimage of a principal down-set is again principal.Equivalently, one component of a Galois connection.

13.17 S• Saturated chain. A chain such that no element can be added between two of its elements without losing theproperty of being totally ordered. If the chain is finite, this means that in every pair of successive elements thelarger one covers the smaller one. See also maximal chain.

• Scattered. A total order is scattered if it has no densely ordered subset.

• Scott-continuous. A monotone function f : P → Q between posets P and Q is Scott-continuous if, for everydirected set D that has a supremum sup D in P, the set {fx | x in D} has the supremum f(sup D) in Q. Stateddifferently, a Scott-continuous function is one that preserves all directed suprema. This is in fact equivalent tobeing continuous with respect to the Scott topology on the respective posets.

• Scott domain. A Scott domain is a partially ordered set which is a bounded complete algebraic cpo.

• Scott open. See Scott topology.

• Scott topology. For a poset P, a subset O is Scott-open if it is an upper set and all directed sets D that have asupremum in O have non-empty intersection with O. The set of all Scott-open sets forms a topology, the Scotttopology.

• Semilattice. A semilattice is a poset in which either all finite non-empty joins (suprema) or all finite non-emptymeets (infima) exist. Accordingly, one speaks of a join-semilattice ormeet-semilattice.

• Smallest element. See least element.

• Sperner property of a partially ordered set

• Sperner poset

• Strictly Sperner poset

• Strongly Sperner poset

13.18. T 123

• Strict order. A strict order is a binary relation that is antisymmetric, transitive, and irreflexive.

• Supremum. For a poset P and a subset X of P, the least element in the set of upper bounds of X (if it exists,which it may not) is called the supremum, join, or least upper bound of X. It is denoted by sup X or

∨X.

The supremum of two elements may be written as sup{x,y} or x ∨ y. If the set X is finite, one speaks of a finitesupremum. The dual notion is called infimum.

• Suzumura consistency. A binary relation R is Suzumura consistent if x R∗ y implies that x R y or not y R x.[1]

• Symmetric. A relation R on a set X is symmetric, if x R y implies y R x, for all elements x, y in X.

13.18 T

• Top. See unit.

• Total order. A total order T is a partial order in which, for each x and y in T, we have x ≤ y or y ≤ x. Totalorders are also called linear orders or chains.

• Total relation. A total or complete relation R on a set X has the property that for all elements x, y of X, atleast one of x R y or y R x holds.

• Transitive. A relation R on a set X is transitive, if x R y and y R z imply x R z, for all elements x, y, z in X.

• Transitive closure. The transitive closure R∗ of a relation R consists of all pairs x,y for which there cists afinite chain x R a, a R b, ..., z R y.[1]

13.19 U

• Unit. The greatest element of a poset P can be called unit or just 1 (if it exists). Another common term forthis element is top. It is the infimum of the empty set and the supremum of P. The dual notion is called zero.

• Up-set. See upper set.

• Upper bound. An upper bound of a subset X of a poset P is an element b of P, such that x ≤ b, for all x in X.The dual notion is called lower bound.

• Upper set. A subset X of a poset P is called an upper set if, for all elements x in X and p in P, x ≤ p impliesthat p is contained in X. The dual notion is called lower set.

13.20 V

• Valuation. Given a lattice X , a valuation ν : X → [0, 1] is strict (i.e., ν(∅) = 0 ), monotone, modular (i.e.,ν(U)+ν(V ) = ν(U ∪V )+ν(U ∩V ) ) and positive. Continuous valuations are a generalization of measures.

13.21 W

• Way-below relation. In a poset P, some element x is way below y, written x<<y, if for all directed subsets Dof P which have a supremum, y ≤ sup D implies x ≤ d for some d in D. One also says that x approximates y.See also domain theory.

• Weak order. A partial order ≤ on a set X is a weak order provided that the poset (X, ≤) is isomorphic to acountable collection of sets ordered by comparison of cardinality.

124 CHAPTER 13. GLOSSARY OF ORDER THEORY

13.22 Z• Zero. The least element of a poset P can be called zero or just 0 (if it exists). Another common term for thiselement is bottom. Zero is the supremum of the empty set and the infimum of P. The dual notion is calledunit.

13.23 Notes[1] Bossert,Walter; Suzumura, Kōtarō (2010). Consistency, choice and rationality. HarvardUniversity Press. ISBN0674052994.

[2] Deng 2008, p. 22

13.24 References

The definitions given here are consistent with those that can be found in the following standard reference books:

• B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge University Press,2002.

• G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, Continuous Lattices andDomains, In Encyclopedia of Mathematics and its Applications, Vol. 93, Cambridge University Press, 2003.

Specific definitions:

• Deng, Bangming (2008), Finite dimensional algebras and quantum groups, Mathematical surveys and mono-graphs 150, American Mathematical Society, ISBN 978-0-8218-4186-0

Chapter 14

Glossary of Riemannian and metricgeometry

“Radius of convexity” redirects here. For the anatomical feature of the radius bone, see Convexity of radius.

This is a glossary of some terms used in Riemannian geometry andmetric geometry— it doesn't cover the terminologyof differential topology.The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expo-sitions of the definitions given below.

• Connection

• Curvature

• Metric space

• Riemannian manifold

See also:

• Glossary of general topology

• Glossary of differential geometry and topology

• List of differential geometry topics

Unless stated otherwise, lettersX, Y, Z below denote metric spaces,M,N denote Riemannian manifolds, |xy| or |xy|Xdenotes the distance between points x and y in X. Italic word denotes a self-reference to this glossary.A caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do nothave exactly the same meaning as in general mathematical usage.Contents :

• Top

• 0–9

• A

• B

• C

• D

125

126 CHAPTER 14. GLOSSARY OF RIEMANNIAN AND METRIC GEOMETRY

• E• F• G• H• I• J• K• L• M• N• O• P• Q• R• S• T• U• V• W• X• Y• Z

14.1 A

Alexandrov space a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the lastone works only in dimension 2)Almost flat manifoldArc-wise isometry the same as path isometry.Autoparallel the same as totally geodesic

14.2 B

Barycenter, see center of mass.bi-Lipschitz map. A map f : X → Y is called bi-Lipschitz if there are positive constants c and C such that for anyx and y in X

c|xy|X ≤ |f(x)f(y)|Y ≤ C|xy|XBusemann function given a ray, γ : [0, ∞)→X, the Busemann function is defined by

Bγ(p) = limt→∞

(|γ(t)− p| − t)

14.3. C 127

14.3 C

Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifoldwith non-positive sectional curvature is diffeomorphic to Rn via the exponential map; for metric spaces, the state-ment that a connected, simply connected complete geodesic metric space with non-positive curvature in the sense ofAlexandrov is a (globally) CAT(0) space.Cartan extended Einstein’s General relativity to Einstein-Cartan theory, using Riemannian-Cartan geometry insteadof Riemannian geometry. This extension provides affine torsion, which allows for non-symmetric curvature tensorsand the incorporation of spin-orbit coupling.Center of mass. A point q ∈ M is called the center of mass of the points p1, p2, . . . , pk if it is a point of globalminimum of the function

f(x) =∑i

|pix|2

Such a point is unique if all distances |pipj | are less than radius of convexity.Christoffel symbolCollapsing manifoldComplete spaceCompletionConformal map is a map which preserves angles.Conformally flat a M is conformally flat if it is locally conformally equivalent to a Euclidean space, for examplestandard sphere is conformally flat.Conjugate points two points p and q on a geodesic γ are called conjugate if there is a Jacobi field on γ which has azero at p and q.Convex function. A function f on a Riemannian manifold is a convex if for any geodesic γ the function f ◦ γ isconvex. A function f is called λ -convex if for any geodesic γ with natural parameter t , the function f ◦ γ(t)− λt2

is convex.Convex A subset K of a Riemannian manifold M is called convex if for any two points in K there is a shortest pathconnecting them which lies entirely in K, see also totally convex.Cotangent bundleCovariant derivativeCut locus

14.4 D

Diameter of a metric space is the supremum of distances between pairs of points.Developable surface is a surface isometric to the plane.Dilation of a map between metric spaces is the infimum of numbers L such that the given map is L-Lipschitz.

14.5 E

Exponential map: Exponential map (Lie theory), Exponential map (Riemannian geometry)

14.6 F

Finsler metric

128 CHAPTER 14. GLOSSARY OF RIEMANNIAN AND METRIC GEOMETRY

First fundamental form for an embedding or immersion is the pullback of the metric tensor.

14.7 G

Geodesic is a curve which locally minimizes distance.Geodesic flow is a flow on a tangent bundle TM of a manifold M, generated by a vector field whose trajectories areof the form (γ(t), γ′(t)) where γ is a geodesic.Gromov-Hausdorff convergenceGeodesic metric space is a metric space where any two points are the endpoints of a minimizing geodesic.

14.8 H

Hadamard space is a complete simply connected space with nonpositive curvature.Horosphere a level set of Busemann function.

14.9 I

Injectivity radius The injectivity radius at a point p of a Riemannian manifold is the largest radius for which theexponential map at p is a diffeomorphism. The injectivity radius of a Riemannian manifold is the infimum of theinjectivity radii at all points. See also cut locus.For complete manifolds, if the injectivity radius at p is a finite number r, then either there is a geodesic of length 2rwhich starts and ends at p or there is a point q conjugate to p (see conjugate point above) and on the distance r fromp. For a closed Riemannian manifold the injectivity radius is either half the minimal length of a closed geodesic orthe minimal distance between conjugate points on a geodesic.Infranilmanifold Given a simply connected nilpotent Lie group N acting on itself by left multiplication and a finitegroup of automorphisms F of N one can define an action of the semidirect productN ⋊F on N. An orbit space of Nby a discrete subgroup ofN ⋊ F which acts freely on N is called an infranilmanifold. An infranilmanifold is finitelycovered by a nilmanifold.Isometry is a map which preserves distances.Intrinsic metric

14.10 J

Jacobi field A Jacobi field is a vector field on a geodesic γ which can be obtained on the following way: Take asmooth one parameter family of geodesics γτ with γ0 = γ , then the Jacobi field is described by

J(t) = ∂γτ (t)/∂τ |τ=0.

Jordan curve

14.11 K

Killing vector field

14.12. L 129

14.12 L

Length metric the same as intrinsic metric.Levi-Civita connection is a natural way to differentiate vector fields on Riemannian manifolds.Lipschitz convergence the convergence defined by Lipschitz metric.Lipschitz distance between metric spaces is the infimum of numbers r such that there is a bijective bi-Lipschitzmapbetween these spaces with constants exp(-r), exp(r).Lipschitz mapLogarithmic map is a right inverse of Exponential map.

14.13 M

Mean curvatureMetric ballMetric tensorMinimal surface is a submanifold with (vector of) mean curvature zero.

14.14 N

Natural parametrization is the parametrization by length.Net. A sub set S of a metric space X is called ϵ -net if for any point in X there is a point in S on the distance ≤ ϵ .This is distinct from topological nets which generalise limits.Nilmanifold: An element of the minimal set of manifolds which includes a point, and has the following property:any oriented S1 -bundle over a nilmanifold is a nilmanifold. It also can be defined as a factor of a connected nilpotentLie group by a lattice.Normal bundle: associated to an imbedding of a manifold M into an ambient Euclidean space RN , the normalbundle is a vector bundle whose fiber at each point p is the orthogonal complement (in RN ) of the tangent spaceTpM .Nonexpanding map same as short map

14.15 P

Parallel transportPolyhedral space a simplicial complex with a metric such that each simplex with induced metric is isometric to asimplex in Euclidean space.Principal curvature is the maximum and minimum normal curvatures at a point on a surface.Principal direction is the direction of the principal curvatures.Path isometryProper metric space is a metric space in which every closed ball is compact. Every proper metric space is complete.

14.16 Q

Quasigeodesic has two meanings; here we give the most common. A map f : R → Y is called quasigeodesic ifthere are constantsK > 0 and C ≥ 0 such that

130 CHAPTER 14. GLOSSARY OF RIEMANNIAN AND METRIC GEOMETRY

1

Kd(x, y)− C ≤ d(f(x), f(y)) ≤ Kd(x, y) + C.

Note that a quasigeodesic is not necessarily a continuous curve.Quasi-isometry. A map f : X → Y is called a quasi-isometry if there are constantsK ≥ 1 and C ≥ 0 such that

1

Kd(x, y)− C ≤ d(f(x), f(y)) ≤ Kd(x, y) + C.

and every point in Y has distance at most C from some point of f(X). Note that a quasi-isometry is not assumed to becontinuous, for example any map between compact metric spaces is a quasi isometry. If there exists a quasi-isometryfrom X to Y, then X and Y are said to be quasi-isometric.

14.17 R

Radius of metric space is the infimum of radii of metric balls which contain the space completely.Radius of convexity at a point p of a Riemannian manifold is the largest radius of a ball which is a convex subset.Ray is a one side infinite geodesic which is minimizing on each intervalRiemann curvature tensorRiemannian manifoldRiemannian submersion is a map between Riemannian manifolds which is submersion and submetry at the sametime.

14.18 S

Second fundamental form is a quadratic form on the tangent space of hypersurface, usually denoted by II, anequivalent way to describe the shape operator of a hypersurface,

II(v, w) = ⟨S(v), w⟩

It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normalspace.Shape operator for a hypersurface M is a linear operator on tangent spaces, Sp: TpM→TpM. If n is a unit normalfield to M and v is a tangent vector then

S(v) = ±∇vn

(there is no standard agreement whether to use + or − in the definition).Short map is a distance non increasing map.Smooth manifoldSol manifold is a factor of a connected solvable Lie group by a lattice.Submetry a short map f between metric spaces is called a submetry if there exists R > 0 such that for any point xand radius r < R we have that image of metric r-ball is an r-ball, i.e.

f(Br(x)) = Br(f(x))

Sub-Riemannian manifoldSystole. The k-systole of M, systk(M) , is the minimal volume of k-cycle nonhomologous to zero.

14.19. T 131

14.19 T

Tangent bundleTotally convex. A subset K of a Riemannian manifold M is called totally convex if for any two points in K anygeodesic connecting them lies entirely in K, see also convex.Totally geodesic submanifold is a submanifold such that all geodesics in the submanifold are also geodesics of thesurrounding manifold.

14.20 U

Uniquely geodesic metric space is a metric space where any two points are the endpoints of a unique minimizinggeodesic.

14.21 W

Word metric on a group is a metric of the Cayley graph constructed using a set of generators.

Chapter 15

Glossary of ring theory

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an additionand a multiplication operation. This is a glossary of some terms of the subject.

15.1 Definition of a ring

ring A ring is a set R with two binary operations, usually called addition (+) and multiplication (×), such that R isan abelian group under addition, R is a monoid under multiplication, and multiplication is both left and rightdistributive over addition. Rings are assumed to have multiplicative identities unless otherwise noted. Theadditive identity is denoted by 0 and the multiplicative identity by 1. (Warning: some books, especially olderbooks, use the term “ring” to mean what here will be called a rng; i.e., they do not require a ring to have amultiplicative identity.)

subring A subset S of the ring (R,+,×) which remains a ring when + and × are restricted to S and contains themultiplicative identity 1 of R is called a subring of R.

15.2 Types of elements

associate In a commutative ring, an element a is called an associate of an element b if a divides b and b divides a.

central An element r of a ring R is central if xr = rx for all x in R. The set of all central elements forms a subring ofR, known as the center of R.

divisor In an integral domain R, an element a is called a divisor of the element b (and we say a divides b) if thereexists an element x in R with ax = b.

idempotent An element r of a ring is idempotent if r2 = r.

integral element For a commutative ring B containing a subring A, an element b is integral over A if it satisfies amonic polynomial with coefficients from A.

irreducible An element x of an integral domain is irreducible if it is not a unit and for any elements a and b suchthat x = ab, either a or b is a unit. Note that every prime element is irreducible, but not necessarily vice versa.

prime element An element x of an integral domain is a prime element if it is not zero and not a unit and wheneverx divides a product ab, x divides a or x divides b.

nilpotent An element r of R is nilpotent if there exists a positive integer n such that rn = 0.

132

15.3. HOMOMORPHISMS AND IDEALS 133

unit or invertible element An element r of the ring R is a unit if there exists an element r−1 such that rr−1 = r−1r= 1. This element r−1 is uniquely determined by r and is called the multiplicative inverse of r. The set of unitsforms a group under multiplication.

von Neumann regular element An element r of a ring R is von Neumann regular if there exists an element x of Rsuch that r = rxr.

zero divisor An element r of R is a left zero divisor if there exists a nonzero element x in R such that rx = 0 and aright zero divisor or if there exists a nonzero element y in R such that yr = 0. An element r of R is a called atwo-sided zero divisor if it is both a left zero divisor and a right zero divisor.

15.3 Homomorphisms and ideals

finitely generated ideal A left ideal I is finitely generated if there exist finitely many elements a1, ..., an such that I= Ra1 + ... + Ran. A right ideal I is finitely generated if there exist finitely many elements a1, ..., an such that I= a1R + ... + anR. A two-sided ideal I is finitely generated if there exist finitely many elements a1, ..., an suchthat I = Ra1R + ... + RanR.

ideal A left ideal I of R is a subgroup of R such that aI ⊆ I for all a ∈ R. A right ideal is a subgroup of R such thatIa ⊆ I for all a ∈ R. An ideal (sometimes called a two-sided ideal for emphasis) is a subgroup which is both aleft ideal and a right ideal.

Jacobson radical The intersection of all maximal left ideals in a ring forms a two-sided ideal, the Jacobson radicalof the ring.

kernel of a ring homomorphism The kernel of a ring homomorphism f : R → S is the set of all elements x of Rsuch that f(x) = 0. Every ideal is the kernel of a ring homomorphism and vice versa.

maximal ideal A left ideal M of the ring R is a maximal left ideal if M ≠ R and the only left ideals containing Mare R and M itself. Maximal right ideals are defined similarly. In commutative rings, there is no difference,and one speaks simply of maximal ideals.

nil ideal An ideal is nil if it consists only of nilpotent elements.

nilpotent ideal An ideal I is nilpotent if the power Ik is {0} for some positive integer k. Every nilpotent ideal is nil,but the converse is not true in general.

nilradical The set of all nilpotent elements in a commutative ring forms an ideal, the nilradical of the ring. Thenilradical is equal to the intersection of all the ring’s prime ideals. It is contained in, but in general not equalto, the ring’s Jacobson radical.

prime ideal An ideal P in a commutative ring R is prime if P ≠ R and if for all a and b in R with ab in P, we havea in P or b in P. Every maximal ideal in a commutative ring is prime. There is also a definition of prime idealfor noncommutative rings.

principal ideal A principal left ideal in a ring R is a left ideal of the form Ra for some element a of R. A principalright ideal is a right ideal of the form aR for some element a of R. A principal ideal is a two-sided ideal of theform RaR for some element a of R.

quotient ring or factor ring Given a ring R and an ideal I of R, the quotient ring is the ring formed by the set R/Iof cosets {a + I : a∈R} together with the operations (a + I) + (b + I) = (a + b) + I and (a + I)(b + I) = ab + I.The relationship between ideals, homomorphisms, and factor rings is summed up in the fundamental theoremon homomorphisms.

134 CHAPTER 15. GLOSSARY OF RING THEORY

radical of an ideal The radical of an ideal I in a commutative ring consists of all those ring elements a power ofwhich lies in I. It is equal to the intersection of all prime ideals containing I.

ring homomorphism A function f : R → S between rings (R, +, ∗) and (S, ⊕, ×) is a ring homomorphism if itsatisfies

f(a + b) = f(a) ⊕ f(b)f(a ∗ b) = f(a) × f(b)f(1) = 1

for all elements a and b of R.

ring monomorphism A ring homomorphism that is injective is a ring monomorphism.

ring isomorphism A ring homomorphism that is bijective is a ring isomorphism. The inverse of a ring isomor-phism is also a ring isomorphism. Two rings are isomorphic if there exists a ring isomorphism between them.Isomorphic rings can be thought as essentially the same, only with different labels on the individual elements.

trivial ideal Every nonzero ring R is guaranteed to have two ideals: the zero ideal and the entire ring R. These idealsare usually referred to as trivial ideals. Right ideals, left ideals, and two-sided ideals other than these are callednontrivial.

15.4 Types of rings

Abelian ring A ring in which all idempotent elements are central is called an Abelian ring. Such rings need not becommutative.

artinian ring A ring satisfying the descending chain condition for left ideals is left artinian; if it satisfies the de-scending chain condition for right ideals, it is right artinian; if it is both left and right artinian, it is calledartinian. Artinian rings are noetherian.

boolean ring A ring in which every element is multiplicatively idempotent is a boolean ring.

commutative ring A ring R is commutative if the multiplication is commutative, i.e. rs = sr for all r,s ∈ R.

Dedekind domain A Dedekind domain is an integral domain in which every ideal has a unique factorization intoprime ideals.

division ring or skew field A ring in which every nonzero element is a unit and 1 ≠ 0 is a division ring.

domain A domain is a nonzero ring with no zero divisors except 0. This is the noncommutative generalization ofintegral domain.

Euclidean domain A Euclidean domain is an integral domain in which a degree function is defined so that “divisionwith remainder” can be carried out. It is so named because the Euclidean algorithm is a well-defined algorithmin these rings. All Euclidean domains are principal ideal domains.

field A field is a commutative division ring. Every finite division ring is a field, as is every finite integral domain.

finitely generated ring a ring that is finitely generated as Z-algebra.

Finitely presented algebra If R is a commutative ring and A is an R-algebra, then A is a finitely presented R-algebra if it is a quotient of a polynomial ring over R in finitely many variables by a finitely generated ideal.[1]

15.4. TYPES OF RINGS 135

hereditary ring A ring is left hereditary if its left ideals are all projective modules. Right hereditary rings are definedanalogously.

integral domain or entire ring A nonzero commutative ring with no zero divisors except 0.

invariant basis number A ring R has invariant basis number if Rm isomorphic to Rn as R-modules implies m = n.

local ring A ring with a unique maximal left ideal is a local ring. These rings also have a unique maximal rightideal, and the left and the right unique maximal ideals coincide. Certain commutative rings can be embeddedin local rings via localization at a prime ideal.

Noetherian ring A ring satisfying the ascending chain condition for left ideals is left Noetherian; a ring satisfyingthe ascending chain condition for right ideals is right Noetherian; a ring that is both left and right Noetherian isNoetherian. A ring is left Noetherian if and only if all its left ideals are finitely generated; analogously for rightNoetherian rings.

null ring See rng of square zero.

perfect ring A left perfect ring is one satisfying the descending chain condition on right principal ideals. They arealso characterized as rings whose flat left modules are all projective modules. Right perfect rings are definedanalogously. Artinian rings are perfect.

prime ring A nonzero ring R is called a prime ring if for any two elements a and b of R with aRb = 0, we haveeither a = 0 or b = 0. This is equivalent to saying that the zero ideal is a prime ideal. Every simple ring andevery domain is a prime ring.

primitive ring A left primitive ring is a ring that has a faithful simple left R-module. Every simple ring is primitive.Primitive rings are prime.

principal ideal domain An integral domain in which every ideal is principal is a principal ideal domain. All prin-cipal ideal domains are unique factorization domains.

quasi-Frobenius ring a special type of Artinian ring which is also a self-injective ring on both sides. Everysemisimple ring is quasi-Frobenius.

rng of square zero A rng in which xy = 0 for all x and y. These are sometimes also called zero rings, even thoughthey usually do not have a 1.

self-injective ring A ring R is left self-injective if the module RR is an injective module. While rings with unity arealways projective as modules, they are not always injective as modules.

semiprimitive ring or Jacobson semisimple ring This is a ring whose Jacobson radical is zero. Von Neumannregular rings and primitive rings are semiprimitive, however quasi-Frobenius rings and local rings are usuallynot semiprimitive.

semisimple ring A semisimple ring is a ring R that has a “nice” decomposition, in the sense that R is a semisimpleleft R-module. Every semisimple ring is also Artinian, and has no nilpotent ideals. The Artin–Wedderburntheorem asserts that every semisimple ring is a finite product of full matrix rings over division rings.

simple ring A non-zero ring which only has trivial two-sided ideals (the zero ideal, the ring itself, and no more) isa simple ring.

trivial ring The ring consisting of a single element 0 = 1, also called the zero ring.

unique factorization domain or factorial ring An integral domain R in which every non-zero non-unit elementcan be written as a product of prime elements of R. This essentially means that every non-zero non-unit can bewritten uniquely as a product of irreducible elements.

136 CHAPTER 15. GLOSSARY OF RING THEORY

von Neumann regular ring A ring for which each element a can be expressed as a = axa for another element x inthe ring. Semisimple rings are von Neumann regular.

zero ring The ring consisting only of a single element 0 = 1, also called the trivial ring. Sometimes “zero ring” isalternatively used to mean rng of square zero.

15.5 Ring constructionsdirect product of a family of rings This is a way to construct a new ring from given rings by taking the cartesian

product of the given rings and defining the algebraic operations component-wise.

endomorphism ring A ring formed by the endomorphisms of an algebraic structure. Usually its multiplication istaken to be function composition, while its addition is pointwise addition of the images.

localization of a ring For commutative rings, a technique to turn a given set of elements of a ring into units. It isnamed Localization because it can be used to make any given ring into a local ring. To localize a ring R, takea multiplicatively closed subset S containing no zero divisors, and formally define their multiplicative inverses,which shall be added intoR. Localization in noncommutative rings is more complicated, and has been in definedseveral different ways.

matrix ring Given a ring R, it is possible to construct matrix rings whose entries come from R. Often these are thesquare matrix rings, but under certain conditions “infinite matrix rings” are also possible. Square matrix ringsarise as endomorphism rings of free modules with finite rank.

opposite ring Given a ring R, its opposite ring Rop has the same underlying set as R, the addition operation is definedas in R, but the product of s and r in Rop is rs, while the product is sr in R.

projective line over a ring Given a ring R, its projective line P(R) provides the context for linear fractional trans-formations of R.

15.5.1 Polynomial rings

Main article: Polynomial ring

differential polynomial ring

formal power series ring

Laurent polynomial ring

monoid ring

polynomial ring GivenR a commutative ring. The polynomial ringR[x] is defined to be the set {anxn+an−1xn−1+

an−2xn−2+. . .+a1x+a0|an, an−1, an−2, . . . , a0 ∈ R}with addition defined by

(∑i aix

i)+(∑

i bixi)=∑

i(ai + bi)xi , and with multiplication defined by

(∑i aix

i)·(∑

j bjxj)=

∑k

(∑i,j:i+j=k aibj

)xk .

Some results about properties of R and R[x]:

• If R is UFD, so is R[x].• If R is Noetherian, so is R[x].

ring of rational functions

skew polynomial ring Given R a ring and an endomorphism σ ∈ End(R) of R. The skew polynomial ring R[x;σ]is defined to be the set {anxn + an−1x

n−1 + an−2xn−2 + . . .+ a1x+ a0|an, an−1, an−2, . . . , a0 ∈ R} ,

with addition defined as usual, and multiplication defined by the relation xa = σ(a)x ∀a ∈ R .

15.6. MISCELLANEOUS 137

15.6 Miscellaneouscharacteristic The characteristic of a ring is the smallest positive integer n satisfying nx = 0 for all elements x of

the ring, if such an n exists. Otherwise, the characteristic is 0.

Krull dimension of a commutative ring The maximal length of a strictly increasing chain of prime ideals in thering.

15.7 Ringlike structures

The following structures include generalizations and other algebraic objects similar to rings.

nearring A structure that is a group under addition, a semigroup under multiplication, and whose multiplicationdistributes on the right over addition.

rng (or pseudo-ring) An algebraic structure satisfying the same properties as a ring, except that multiplication neednot have an identity element. The term “rng” is meant to suggest that it is a “ring” without an "identity”.

semiring An algebraic structure satisfying the same properties as a ring, except that addition need only be an abelianmonoid operation, rather than an abelian group operation. That is, elements in a semiring need not have additiveinverses.

15.8 See also• Glossary of field theory

• Glossary of module theory

• Glossary of commutative algebra

15.9 Notes[1] Grothendieck & Dieudonné 1964, §1.4.1

15.10 References• Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude localedes schémas et des morphismes de schémas, Première partie”. Publications Mathématiques de l'IHÉS 20.doi:10.1007/bf02684747. MR 0173675.

Chapter 16

Glossary of semisimple groups

This is a glossary for the terminology applied in the mathematical theories of semisimple Lie groups. It also coversterms related to their Lie algebras, their representation theory, and various geometric, algebraic and combinatorialstructures that occur in connection with the development of what is a central theory of contemporary mathematics.Contents :

• Top

• 0–9

• A

• B

• C

• D

• E

• F

• G

• H

• I

• J

• K

• L

• M

• N

• O

• P

• Q

• R

• S

• T

138

16.1. A 139

• U

• V

• W

• X

• Y

• Z

16.1 A• Adjoint representation

The adjoint representation of any Lie group is its action on its Lie algebra, derived from the conjugation action of thegroup on itself.

• Affine Lie algebra

An affine Lie algebra is a particular type of Kac–Moody algebra.

• Algebraic group

16.2 B• (B, N) pair

• Borel subgroup

• Borel-Bott-Weil theorem

• Bruhat decomposition

16.3 C• Cartan decomposition

• Cartan matrix

• Cartan subalgebra

• Cartan subgroup

• Casimir invariant

• Clebsch–Gordan coefficients

• Compact Lie group

• Complex reflection group

• coroot

• Coxeter group

• Coxeter number

• Cuspidal representation

140 CHAPTER 16. GLOSSARY OF SEMISIMPLE GROUPS

16.4 D• Discrete series

• Dominant weight

The irreducible representations of a simply-connected compact Lie group are indexed by their highest weight. Thesedominant weights form the lattice points in an orthant in the weight lattice of the Lie group.

• Dynkin diagram

16.5 E• E6 (mathematics)

• E7 (mathematics)

• E7½ (Lie algebra)

• E8 (mathematics)

• En (Lie algebra)

• Exceptional Lie algebra

16.6 F• F4 (mathematics)

• Flag manifold

• Fundamental representation

For the irreducible representations of a simply-connected compact Lie group there exists a set of fundamental weights,indexed by the vertices of the Dynkin diagram of G, such that dominant weights are simply non-negative integer linearcombinations of the fundamental weights.The corresponding irreducible representations are the fundamental representations of the Lie group. In particular,from the expansion of a dominant weight in terms of the fundamental weights, one can take a corresponding tensorproduct of the fundamental representations and extract one copy of the irreducible representation corresponding tothat dominant weight.In the case of the special unitary group SU(n), the n − 1 fundamental representations are the wedge products

Altk Cn

consisting of alternating tensors, for k=1,2,...,n-1.

• Fundamental Weyl chamber

16.7 G• G2 (mathematics)

• Generalized Cartan matrix

• Generalized Kac–Moody algebra

• Generalized Verma module

16.8. H 141

16.8 H

• Harish-Chandra homomorphism

• Highest weight

• Highest weight module

16.9 I

• Iwasawa decomposition

16.10 J

16.11 K

• Kac–Moody algebra

• Killing form

• Kirillov character formula

16.12 L

• Langlands decomposition

• Langlands dual

• Levi decomposition

• Lie algebra

• Lowest weight

16.13 M

• Maximal compact subgroup

• Maximal torus

16.14 N

• Nilpotent cone

Elements in a semisimple Lie algebra that are represented in every linear representation by a nilpotent endomorphism.

142 CHAPTER 16. GLOSSARY OF SEMISIMPLE GROUPS

16.15 O

16.16 P

• Parabolic subgroup

• Peter–Weyl theorem

• Positive root

16.17 Q

16.18 R

• Real form

• Reductive group

• Reflection group

• Root datum

• Root system

16.19 S

• Schur polynomial

A Schur polynomial is a symmetric function, of a type occurring in the Weyl character formula applied to unitarygroups.

• Semisimple Lie algebra

• Semisimple Lie group

• Simple Lie algebra

• Simple Lie group

• Simple root

• Simply laced group

A simple Lie group is simply laced when its Dynkin diagram is without multiple edges

• Steinberg representation

16.20 T

16.21 U

• Unitarian trick

16.22. V 143

16.22 V• Verma module

16.23 W• Weight (representation theory)

• Weight module

• Weight space

• Weyl chamber

A Weyl chamber is one of the connected components of the complement in V, a real vector space on which a rootsystem is defined, when the hyperplanes orthogonal to the root vectors are removed.

• Weyl character formula

The Weyl character formula gives in closed form the characters of the irreducible complex representations of thesimple Lie groups.

• Weyl group

Chapter 17

Glossary of shapes with metaphoricalnames

Many shapes have metaphorical names, i.e., their names are metaphors: these shapes are named after a mostcommon object that has it. For example, “U-shape” is a shape that resembles the letter U, a bell-shaped curve has theshape of the vertical cross-section of a bell, etc.These terms may variously refer to objects, their cross sections or projections.Some of these names are “classical terms”, i.e., words of Latin or Ancient Greek etymology. Others are Englishlanguage constructs (although the base words may have non-English etymology). In some disciplines, where shapesof subjects in question are a very important consideration, the shape naming may be quite elaborate, see, e.g., thetaxonomy of shapes of plant leaves in botany.

• Astroid

• Bell-shaped curve

• Biconic shape, a shape in a way opposite to the hourglass: it is based on two oppositely oriented cones ortruncated cones with their bases joined; the cones are not necessarily the same

• Bowtie shape, in two dimensions• Atmospheric reentry apparatus• Centerbody of an inlet cone in ramjets

• Bow shape

• Bow curve

• Bullet Nose[1] an open-ended hourglass

• Butterfly curve[2]

• Cocked Hat curve, also known as Bicorn[3]

• Cone (from the Greek word for « pine cone »)

• Egg-shaped, see “Oval”, below

• Fish bladder or Lens shape (the latter taking its name from the shape of the lentil seed)

• Geoid (From Greek Ge (γη) for "Earth"), the term specifically introduced to denote the approximation of theshape of the Earth, which is approximately spherical, but not exactly so

• Heart shape, long been used for its varied symbolism

• Hourglass shape or hourglass figure, the one that resembles an hourglass; nearly symmetric shape wide at itsends and narrow in the middle; some flat shapes may be alternatively compared to the figure eight or hourglass

144

17.1. NUMBERS AND LETTERS 145

• Dog bone shape, an hourglass with rounded ends[4]

• Hourglass corset• Ntama• Hourglass Nebula

• Inverted bell

• Lune, from the Latin word for the Moon

• Maltese Cross curve[5]

• Mushroom shape, which became infamous as a result of the mushroom cloud

• Oval (from the Latin “ovum” for egg), a descriptive term applied to several kinds of “rounded” shapes, includ-ing the egg shape

• Pear shaped, in reference to the shape of a pear, i.e., a generally rounded shape, tapered towards the top andmore spherical/circular at the bottom

• Rod, a 3-dimensional, solid (filled) cylinder

• Rod shaped bacteria

• Scarabaeus curve[6] resembling a scarab

• serpentine, shaped like a snake

• Stadium, two half-circles joined by straight sides[7]

• Stirrup curve[8]

• Star a figure with multiple sharp points

• Sunburst

• Tomahawk

17.1 Numbers and letters• Figure 8, the shape that resembles the numeral 8, symbol for infinity when laid on its side

• A-shape, the shape that resembles the capital letter A

• A-Frame, the shape of a common structure that resembles the capital letter A• A-Frame house, a common style of house construction• A-shirt

• C-shape, the shape that resembles the capital letter C

• D-shape, the shape that resembles the capital letter D

• D-ring

• Deltoid, the shape that resembles the Greek capital letter Δ

• Deltahedron• Deltoid muscle• River delta

• E-shape, the shape that resembles the capital letter E

• Magnetic cores of transformers may be E-shaped• A number of notable buildings have an E-shaped floorplan

146 CHAPTER 17. GLOSSARY OF SHAPES WITH METAPHORICAL NAMES

• H-shape, the shape that resembles the capital letter H

• H-beam, a beam with H-shaped section• Goals in several sports (gridiron football (old style), Gaelic football, rugby, hurling) are described as“H-shaped”

• I-shape, the shape that resembles the capital letter I in a serif font, i.e., with horizontal strokes

• I-beam, a beam with an I-shaped section• The court in the Mesoamerican ballgame is I-shaped

• J-shape, the shape that resembles the capital letter J

• K-shape, the shape that resembles the capital letter K

• K Turn

• L-shape, the shape that resembles the capital letter L

• The L-Shaped Room• L game• L-shaped recession

• N-shape, the shape that resembles the capital letter N (interchangeable with the Z-shape)

• O-shape, the shape that resembles the capital letter O

• O-ring

• P-shape, the shape that resembles the capital letter P

• P-trap, a P-shaped pipe under a sink or basin

• S-shape, the shape that resembles the capital letter S

• The sigmoid colon, an S-shaped bend in the human intestine

• T-shape, the shape that resembles the capital letter T

• T junction• T-shaped (chemistry)• T-shaped skills, a format for résumés• T-shirt

• U-shape, the shape that resembles the capital letter U

• U-shaped valley• U-turn• U-shaped recession

• Hyoid, the shape that resembles the Greek letter υ

• Hyoid bone

• V-shape, the shape that resembles the letter V, also known as the Chevron (which includes the inverted-Vshape)

• V-shaped valley• V-shaped recession• V-shaped body – male human body shape with broad shoulders• V-shaped passage grave• V sign

17.2. SEE ALSO 147

• W-shape, the shape that resembles the capital letter W

• W-shaped recession

• X-shape, the shape that resembles the letter X

• Saltire

• Chiasm, crossings that resemble the Greek letter Χ

• Chiasmus• Chiastic structure• Optic chiasm

• Y-shape, the shape that resembles the letter Y

• Y-front briefs

17.2 See also• List of geometric shapes

• The Category:Curves lists numerous metaphorical names, such as

• Bean curves, also called Nephroids, from the Greek word for kidney

17.3 References[1] “Bullet Nose – from Wolfram MathWorld”. Mathworld.wolfram.com. 2013-05-08. Retrieved 2013-05-20.

[2] “Butterfly Curve – from Wolfram MathWorld”. Mathworld.wolfram.com. 2013-05-08. Retrieved 2013-05-20.

[3] “Bicorn – from Wolfram MathWorld”. Mathworld.wolfram.com. 2013-05-08. Retrieved 2013-05-20.

[4] “Cassini Ovals – from Wolfram MathWorld”. Mathworld.wolfram.com. 2013-05-08. Retrieved 2013-05-20.

[5] “Maltese Cross Curve – from Wolfram MathWorld”. Mathworld.wolfram.com. 2013-05-08. Retrieved 2013-05-20.

[6] “Scarabaeus – from Wolfram MathWorld”. Mathworld.wolfram.com. 2013-05-08. Retrieved 2013-05-20.

[7] “Stadium – from Wolfram MathWorld”. Mathworld.wolfram.com. 2013-05-08. Retrieved 2013-05-20.

[8] “Stirrup Curve – from Wolfram MathWorld”. Mathworld.wolfram.com. 2013-05-08. Retrieved 2013-05-20.

Chapter 18

Glossary of string theory

This page is a glossary of terms in string theory, including related areas such as supergravity, supersymmetry, andhigh energy physics.Contents :

• Conventions

• αβγ

• !$@

• A

• B

• C

• D

• E

• F

• G

• H

• I

• J

• K

• L

• M

• N

• O

• P

• Q

• R

• S

• T

148

18.1. CONVENTIONS 149

• U

• V

• W

• XYZ

•

•

• See also

• References

18.1 Conventions-bein A suffix indicating a frame, where the first part is a German word indicating the dimension (as in zweibein,

vierbein, and so on).

-ino The superpartners of bosons are often denoted by the suffix -ino; for example, photon/photino.

s- The superpartners of fermions are often denoted by adding s- at the beginning; for example, quark/squark.

18.2 αβγα 1. Fine-structure constant

2. Regge slope, or inverse of the string tension

3. A Fourier coefficient of a spacetime coordinate.

4. αs is the strong coupling constant

β 1. One of the two conformal superghost fields β, γ used in the BRST quantization of the superstring

2. Euler beta function

3. Beta function describing the change of coupling constant under the renormalization group flow

γ 1. Dirac matrix

2. One of the two conformal superghost fields β, γ used in the BRST quantization of the superstring

3. World-sheet metric γab(σ,τ)

4. Photon

5. Euler constant .57721...

Γ 1. Lattice

2. Euler Gamma function

3. Dirac matrix

4. Width of some scattering process

δ 1. Kronecker delta function

2. An infinitesimal change in something; for example δL is an infinitesimal change in L

Δ 1. Propagator

150 CHAPTER 18. GLOSSARY OF STRING THEORY

2. Delta baryon, a baryon with 3 light quarks and isospin 3/2

3. Laplace operator in Euclidean space or more generally a Riemannian manifold

ε 1. Small positive real number

2. Antisymmetric tensor

η 1. Flat Lorentzian metric on spacetime

2. Dedekind eta function, a weight 1/2 modular form

3. Neutral flavor meson with PC= –+

θ 1. Theta function

2. θ is the Cabbibo angle

3. θ is the Weinberg angle, also called the weak mixing angle

Λ Cosmological constant

2. Large energy or large mass cutoff in regularization

3. Lambda baryon, a baryon with 2 light quarks and isospin 0

μ 1. Renormalization scale, with the dimensions of mass

2. Muon

ν Neutrino

Ξ 1. Xi baryon, a baryon with 1 light quark

π 1. 3.14159...

2. Pion

Π The momentum density conjugate to X

ρ Rho meson, a light meson with PC= ––

σ 1. Spacelike coordinate on the world-sheet

2. Scattering cross section

3. Pauli matrix

4. See #sigma model

Σ 1. Sigma baryon, a baryon with 2 light quarks and isospin 1

τ 1. Timelike coordinate on the world-sheet

2. Element of the upper half plane

3. Tauon

Υ Upsilon meson (bb)

φ Scalar field

χ Neutral-flavor heavy meson with PC= ++

ψ 1. Spinor field

2. Psi meson (cc)

Ω 1. Density of something in the universe; for example, Ων is the neutrino density

2. Omega baryon, a baryon with no light quarks

18.3. !$@ 151

18.3 !$@

' (prime) X′ means ∂X/∂σ.

dot above letter Ẋ means ∂X/∂τ

∇ 1. A covariant derivative

2. The del operator.

□ The D'Alembert operator, or non-Euclidean Laplacian.

[,] A commutator: [A,B] = AB–BA.

{,} An anticommutator: {A,B} = AB+BA.

18.4 A

A 1. A connection 1-form

2. Short for antiperiodic, a boundary condition on strings.

3. Short for axial vector

4. An asymmetry

action Main article: action (physics)A function S on the space of fields given (formally) by the integral of the Lagrangian density over spacetime,whose stationary points are the solutions of the equations of motion.

ADE Main article: ADE classificationRefers to the ADE classification (An,D , E6, E7, E8) of simply laced Dynkin diagrams, and to several relatedclassifications of Lie algebras, singularities and so on.

ADHM Initials of Atiyah, Drinfeld, Hitchin, and Manin, as in the ADHM construction of instantons.

ADM Initials of Arnowitt, Deser, and Misner, as in ADM energy, a way of defining the global energy in an asymp-totically flat spacetime, or ADM decomposition of a metric, or ADM formalism.

AdS Anti-de Sitter, as in anti-de Sitter space, a Lorentzian analogue of hyperbolic space

AdS/CFT Anti-de Sitter/conformal field theory, especially the AdS/CFT correspondence.

ALE Asymptotically locally Euclidean

ALEPH ALEPH experiment at LEP

AMSB Anomaly mediation supersymmetry breaking

ASD Anti self-dual (connection)

ATLAS The ATLAS experiment at CERN, a particle detector.

axino Main article: AxinoA hypothetical supersymmetric partner of an axion.

axion Main article: AxionA hypothetical scalar particle whose mass arises from a coupling rather than from a mass term in the La-grangian, used to resolve the strong CP problem.

152 CHAPTER 18. GLOSSARY OF STRING THEORY

18.5 Bb 1. One of the two conformal ghost fields b, c used in the BRST quantization of the bosonic string.

2. A bottom quark.

B 1. Baryon number

2. Short for boson.

3. Short for baryon.

4. Short for backward;for example, σB is the cross section for backward scattering.

5. a bottom meson.

BAO Baryon acoustic oscillation

BB Big Bang

BBN Big Bang nucleosynthesis

bino Main article: gauginoA hypothetical supersymmetric partner of the gauge field corresponding to weak hypercharge.

BIon A BPS solution representing an infinite string ending on a D-brane. Named after the Born–Infeld action.

BPS Main article: BPS stateA state related to the Bogomol'nyi–Prasad–Sommerfield bound.

BR Branching ratio

BRS

BRST quantization Main article: BRST quantizationShort for Becchi, Rouet, Stora and Tyutin, who introduced the BRST quantization of gauge theories.

brane Short for membrane. a higher-dimensional manifold moving in spacetime. See also p-brane, D-brane.

BTZ Initials of Bañados–Teitelboim–Zanelli, as in BTZ black hole, a black hole in 2+1-dimensional gravity.

BV Batalin–Vilkovisky, as in Batalin–Vilkovisky formalism.

18.6 Cc 1. The speed of light, when not using units where this is 1.

2. A central charge of the Virasoro algebra or similar algebra.

3. One of the two conformal ghost fields b, c used in the BRST quantization of the bosonic string.

4. A Chern class.

5. A charm quark.

C 1. Charge, especially the charge symmetry.

Calabi–Yau Main article: Calabi–Yau manifoldA Kähler manifold with vanishing Ricci curvature, used for compactifying string theories.

CAR Canonical anticommutation relations

CBR Cosmic background radiation

CC 1. Charged current (weak interaction).

2. Complex conjugate

18.6. C 153

3. Compatibility condition

CCR Canonical commutation relation

CCR and CAR algebras

CDF Collider Detector at Fermilab

CDM Cold dark matter

CERN Conseil Européen pour la Recherche Nucléaire

chargino Main article: charginoA hypothetical charged supersymmetric partner of a gauge boson.

Chern–Simons 1.Main article: Chern–Simons theory

2.Main article: Chern–Simons form

chiral 1. Not invariant under the parity symmetry. The word comes from the Greek χειρ meaning “hand"; the terms“left-handed” and “right-handed” are often used to describe chiral objects.Main article: Chirality (physics)

2. A chiral multiplet is a type of supermutliplet of a supersymmetry algebra.

CIPT Contour improved perturbation theory

CKG Short for conformal Killing group.

CKM The Cabibbo–Kobayashi–Maskawa matrix.

CKS Short for conformal Killing spinor.

CKV Short for conformal Killing vector.

CFT Conformal field theory

Chan–Paton A Chan–Paton charge is a degree of freedom carried by an open string on its endpoints.

cl 1. Short for classical (for example, Scl is the classical action).

2. CL is short for confidence limit.

closed A closed string is one with no ends.

CM Center of mass (frame)

CMB

CMBR Cosmic microwave background radiation

CMS 1. The Compact Muon Solenoid at CERN, a particle detector.

2. Short for the Center-of-Momentum System, a coordinate system where the total momentum is 0.

compactification A method for reducing the apparent dimension of spacetime by wrapping the string around acompact manifold.

cosmological constant Main article: cosmological constantThe constant term of the Lagrangian, inducing a term in the action proportional to the volume of spacetime

CP Short for Charge–Parity, as in CP symmetry.

CPC Short for Charge–Parity conservation.

154 CHAPTER 18. GLOSSARY OF STRING THEORY

CPT Short for Charge–Parity–Time, as in CPT symmetry or CPT theorem.

CPV Short for Charge–Parity violation.

critical The critical dimension is the spacetime dimension in which a string or superstring theory is consistent;usually 26 for string theories and 10 for superstring theories.

CVC Conserved vector current.

CY Short for Calabi–Yau, as in Calabi–Yau manifold, a Ricci-flat Kähler manifold, often used for compactifyingsuperstring theories.

18.7 Dd 1. The exterior derivative of a form.

2. A down quark.

3. The dimension of spacetime.

D 1. Short for Dirichlet, as in D-brane

2. The dimension of spacetime

3. A connection or differential operator

4. A Dynkin diagram of an orthogonal group in even dimensions.

5. A charmed meson.

D0 Main article: DØ experiment

D-brane

Dp-brane Main article: D-braneShort for Dirichlet (mem)brane, a submanifold (of dimension p+1) on which the ends of strings are constrainedto lie, so that the strings satisfy Dirichlet boundary conditions.

D-string A D1-brane

DBI Short for Dirac–Born–Infeld, as in the DBI action, an action based on the Born–Infeld action, a modificationof the Maxwell action of electrodynamics.

DDF Initials of Del Guidice, Di Vecchia, and Fubini, as in Del Guidice–Di Vecchia–Fubini operator, operatorsgenerating an oscillator algebra.

DELPHI DELPHI experiment at LEP.

DESY Deutsches Elektronen-Synchrotron

DGLAP Initials of Dokshitzer–Gribov–Lipatov–Altarelli–Parisi who introduced the DGLAP evolution equation inQCD.

Diff Diffeomorphism or diffeomorphism group.

dilatino Main article: dilatinoA supersymmetric partner of the dilaton.

dilaton Main article: dilatonA massless scalar particle, related to dilations of spacetime.

Dirichlet Dirichlet boundary conditions on an open string say that the ends of the string are fixed (often lying on aD-brane).

DIS Deep inelastic scattering

18.8. E 155

DLCQ Discrete light-cone quantization

DM Dark matter

DØ Main article: DØ experiment

Dp-brane Main article: D-braneShort for Dirichlet (mem)brane, a submanifold (of dimension p+1) on which the ends of strings are constrainedto lie, so that the strings satisfy Dirichlet boundary conditions.

DR 1. Short for dimensional regularization.

2. Short for dimensional reduction, a way of constructing theories from simpler theories in higher dimensions,sometimes by making fields invariant under some spacelike translations.

dS de Sitter, as in de Sitter space, a Lorentzian analogue of a sphere

dS/CFT de Sitter/conformal field theory, especially the dS/CFT correspondence.

dual resonance model Main article: Dual resonance modelAn early precursor of string theory.

duality Main article: String dualityA hidden connection between two different theories, such as S-duality, T-duality, U-duality, mysterious duality.

DY Initials of Drell–Yan, as in DY process.

dyon Main article: DyonA hypothetical particle with both electrical and magnetic charge.

18.8 E

e 1. Euler’s constant

2. A frame

3. An electron

E Energy

E₆ Main article: E6 (mathematics)The exceptional Lie algebra of rank 6 and dimension 78.

E₇ Main article: E7 (mathematics)The exceptional Lie algebra of rank 7 and dimension 133.

E₈ Main article: E8 (mathematics)The exceptional Lie algebra of rank 8 and dimension 248.

eff Short for effective (field theory).

EFT Effective field theory, a low-energy approximation to a theory.

einbein A frame in 1 dimension

elfbein A frame in 11 dimensions

energy–momentum tensor Main article: Energy–momentum tensorA symmetric tensor T (also called the stress-energy tensor) describing the variation of the action under changesin the metric, whose components give the local energy, momentum and stress densities. In flat spacetimes itcan also be given by combining the Noether currents of the translation symmetries.

EWSB Electro-weak symmetry breaking.

156 CHAPTER 18. GLOSSARY OF STRING THEORY

18.9 F

F 1. A curvature form of a connection

2. The world-sheet fermion number.

3. Short for fermion

3. Short for forward;for example, σF is the cross section for backward scattering.

F₄ Main article: F4 (mathematics)The exceptional Lie algebra of rank 4 and dimension 52.

FCNC Flavor-changing neutral current.

field A section of a fiber bundle

FOPT Fixed-order perturbation theory.

F-string Fundamental string

F-theory Main article: F-theoryPossibly an abbreviation of father theory. A 12-dimensional string theory introduced by Vafa.

FRW Friedman–Robertson–Walker metric on spacetime

18.10 G

g 1. A metric

2. A coupling constant

3. The genus of a Riemann surface.

4. A gluon.

G 1. Newton’s gravitational constant, sometimes written GN.

2. The Fermi coupling constant for weak interactions, sometimes written GF.

3. Gn is an odd element of the Ramond or Neveu–Schwarz superalgebra.

G₂ Main article: G2 (mathematics)The exceptional Lie algebra of rank 2 and dimension 14, or a G2 manifold with G₂ holonomy.

gaugino A spin 1/2 supersymmetric partner of a gauge boson.

gh Abbreviation for ghost; for example, Sgh is the ghost action.

ghost A vector of negative norm.

GKO Short for Goddard–Kent–Olive. The GKO construction, also called the coset construction, is a way of con-structing unitary discrete series representations of the Virasoro algebra.

GL A general linear group.

gluino Main article: gluinoA hypothetical supersymmetric partner of a gluon.

gluon Main article: gluonA gauge boson associated with the strong force.

GMSB Gauge mediated supersymmetry breaking.

18.11. H 157

goldstino Main article: goldstinoA massless spin 1/2 particle associated with spontaneous breakdown of supersymmetry, analogous to theGoldstone boson.

GR General relativity

graviton Main article: gravitonA conjectural spin 2 massless particle responsible for gravity.

gravitino Main article: gravitinoA supersymmetric partner of the graviton.

Green Named for Michael Green.

GS Green–Schwarz formalism, a way of incorporating supersymmetry into string theory that is supersymmetric in10-dimensional spacetime.

GSO Short for Ferdinando Gliozzi, Joël Scherk, and David A. Olive, as in the GSO projection, a projection insuperstring theory that eliminates tachyons.

GSW The 2-volume work on superstring theory by Green, Schwarz, and Witten.

GUT Grand unified theory, a hypothetical theory unifying the strong and electroweak forces.

GWS Glashow–Weinberg–Salem theory of the electroweak force.

GZK The Greisen–Zatsepin–Kuzmin limit on the energy of cosmic background radiation from distant sources.

18.11 H

h 1. The weight of a field (for example, its eigenvalue for L0).

2. Hermitian; for example, h.c. stands form hermitian conjugate.

H 1. The Hamiltonian.

2. The Higgs boson.

3. The Hubble constant.

Haag–Lopuszanski–Sohnius theorem Main article: Haag–Lopuszanski–Sohnius theoremA theorem describing the possible supersymmetries of a quantum field theory, generalizing the Coleman–Mandula theorem.

Hagedorn temperature Main article: Hagedorn temperatureThe temperature above which the partition function diverges due to the exponentially increasing number ofstring states.

h.c.

hc Hermitian conjugate

HCMS Hadronic center of mass (frame)

HDM Higgs doublet model

HE Short for heterotic-E28, a heterotic string theory based on the group E28.

helicity Main article: Helicity (particle physics)The projection of the spin of a massless particle in the direction of its momentum.

HERA Hadron Elektron Ring Anlage

158 CHAPTER 18. GLOSSARY OF STRING THEORY

heterotic Main article: heterotic stringNamed after the Greek word heterosis, meaning hybrid vigour. A hybrid of bosonic string theory and super-string theory, introduced by David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm in 1985.

Higgs boson Main article: Higgs bosonA massive scalar particle related to the spontaneous symmetry breaking mechanism in the electroweak theory.

Higgsino Main article: HiggsinoA hypothetical supersymmetric partner of a Higgs boson.

HO Short for heterotic-orthogonal, a heterotic string theory based on the orthogonal group O32(R).

holographic principle Main article: Holographic principle

HQET Heavy quark effective theory

Hyperkähler

Hyperkaehler Main article: Hyperkähler manifoldA Riemannian manifold with holonomy contained in the compact form of the symplectic group.

Hypermultiplet Main article: HypermultipletA type of supermultiplet (representation) of an extended supersymmetry algebra.

18.12 I

i √–1

I Isospin.

IGM Intergalactic medium

inflation Main article: Inflation (cosmology)A hypothetical very rapid increase in the size of the very early universe.

instanton Main article: InstantonA self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold.

int Short for interaction; for example, H ᵢ might be an interaction Hamiltonian.

inv Short for invisible; for example, Γᵢ ᵥ is the width for invisible decays (those unobseverd by an experiment).

18.13 J

J 1. A current

2. A source

3. Spin.

18.14 K

k A momentum

K A kaon (a strange meson).

K3 Main article: K3 surfaceA simply connected compact complex surface of Kodaira dimension 0

18.15. L 159

K-theory Main articles: K-theory and K-theory (physics)A cohomology theory based on vector bundles.

Kac–Moody algebra Main article: Kac–Moody algebraA central extension of a loop algebra.

KählerKaehler Named after Erich Kähler1. A Kähler manifold is a complex manifold with a compatible Riemannian metric.2. A Kähler metric is the metric on a Kähler manifold.3. A Kähler potential is a function of superfields used to construct a Lagrangian.Kalb–Ramond field Main article: Kalb–Ramond field

KK Kaluza–KleinKM 1. The Kobayashi–Maskawa mechanism for CP violation.2. Kac–Moody algebra.KZ Initials of Knizhnik and Zamolodchikov, as in KZ equation, a differential equation related to the primary fields

of a current algebra.

18.15 LL 1. A Lagrangian2. Ln is an element of the Virasoro algebra.3. An abbreviation for left (moving modes).4. Lepton number5. Short for leptonL3 L3 experiment at LEP.Lagrangian (field theory) Main article: Lagrangian (field theory)

A function on the jet space of a fiber bundle.landscape Main article: String theory landscape

The (conjectural) moduli space of all (vacuums of) string theories.LEP The Large Electron–Positron Collider at CERN.lepton Main article: lepton

A elementary particle of spin 1/2 that is unaffected by the strong force.LH Left-handedLHC The large hadron collider at CERN.little string theory Main article: Little string theory

LL Double logarithmicLO Leading order (term)LQG Loop quantum gravityLQC Loop quantum cosmologyLSP Abbreviation for lightest supersymmetric particle.LSS Large scale structure (of the universe).

160 CHAPTER 18. GLOSSARY OF STRING THEORY

18.16 Mm A mass of a fermion. For example, mt is the mass of the top quark t.

M The mass of a boson; for example, MZ is the mass of the Z-boson.

Majorana fermion

Majorana spinor Main articles: Majorana fermion and Majorana spinorA fermion or spinor with a reality condition, in spacetimes of dimension 2, 3, 4 mod 8.

Majorana–Weyl fermion

Majorana–Weyl spinor Main articles: Majorana–Weyl fermion and Majorana–Weyl spinorA half-spinor with a reality condition, in spacetimes of dimension 2 mod 8.

Mandelstam variable Main article: Mandelstam variablesA sum or difference of two of the four incoming or outgoing momenta of a 2-particle interaction.

matrix theory

M(atrix) theory Main article: Matrix string theoryOne of several non-perturbative formulations of string theory or M-theory using infinite matrices.

M-brane

membrane Main article: Membrane (M-theory)A higher dimensional analogue of a string.

MC Monte Carlo integration

MCG Main article: Mapping class group

minimal model Certain solvable conformal field theories.

Mirror symmetry (string theory) Main article: Mirror symmetry (string theory)A partly conjectural relation between a type IIA superstring theory compactified on a Calabi–Yau manifoldand a type IIB superstring theory compactified on a different “mirror” Calabi–Yau manifold.

MLLA Modified leading logarithm approximation.

MNS Maki–Nakagawa–Sakata matrix for neutrino mixing

monopole Main articles: Magnetic monopole, Dirac monopole, Wu–Yang monopole, 't Hooft–Polyakov monopole,Seiberg-Witten monopole and Monopole (mathematics)A hypothetical particle similar to a “magnet with only one pole”.

Montonen–Olive duality Main article: Montonen–Olive dualityAn early case of S-duality.

MS minimal subtraction (a renormalization scheme). MS is the modified minimal subtraction scheme.

MSM Abbreviation for minimal standard model.

MSSM Abbreviation for minimal supersymmetric standard model.

mSUGRA Minimal model of supergravity.

M-theory Main article: M-theoryAn 11-dimensional theory introduced in the second string theory revolution to unify the 5 known superstringtheories. The letter M has been said to stand for membrane, matrix, magic, mystery, monster, and so on.

MSW Mikheyev–Smirnov–Wolfenstein effect concerning neutrino oscillations in matter.

multiplet A linear representation of a Lie algebra or group.

A collection of elementary particles corresponding to a basis of a representation.

18.17. N 161

18.17 N

N 1. The number of times each irreducible real spinor representation appears in the fermionic part of a supersymmetryalgebra or super Minkowski space. It is often used in the description of an extended supersymmetry algebra,as in N=2 superconformal algebra and so on.

2. A nucleon, a baryon with 3 light quarks and isospin 1/2 (such as a proton or neutron).

3. The number of some type of particle.

Nambu–Goto action Main article: Nambu–Goto actionAn action for strings, proportional to the area of the worldsheet.

NC Neutral current (weak interaction).

Neumann Neumann boundary conditions on an open string say that the momentum normal to the boundary of theworld-sheet is zero.

neutralino Main article: NeutralinoA hypothetical supersymmetric partner of a gauge boson with zero charge.

Neveu Named for André Neveu.

Neveu–Schwarz algebra Main article: Neveu–Schwarz algebraA supersymmetric extension of the Virasoro algebra, similar to the Ramond algebra.

NG 1. Short for Nambu–Goto, as in Nambu–Goto action.

2. Short for Nambu–Goldstone, as in Nambu–Goldstone boson.

NLL Next to leading logarithmic (term).

NLO Next to leading order (term).

NLSP next-to-lightest sypersymmetric particle

NMSSM Next-to-Minimal Supersymmetric Standard Model.

NNLL Next to next to leading logarithmic (term).

NNLO Next to next to leading order (term).

NNNLL Next to next to next to leading logarithmic (term).

no-ghost theorem Main article: No-ghost theoremA theorem stating that some hermitian form is positive semidefinite, in other words has no ghosts (negativenorm vectors). The name is a word-play on no-go theorem.

NR Non-relativistic

NRQCD Non-relativistic quantum chromodynamics

NS Neveu–Schwarz, especially the Neveu–Schwarz algebra

NS–NS A sector with Neveu–Schwarz conditions on left and right moving modes.

NS–R A sector with Neveu–Schwarz conditions on left moving modes and Ramond conditions on right movingmodes.

NUT The initials of E. Newman, L. Tamburino, and T. Unti, mainly used in Taub–NUT vacuum, a solution toEinsteins’ equations.

162 CHAPTER 18. GLOSSARY OF STRING THEORY

18.18 OO An orthogonal groupOCQ Short for old covariant quantizationOPAL The OPAL detector at LEP.open An open string is one with two ends.OPEoperator product expansion Main article: Operator product expansion

A description of short-distance singularities of fields.orbifold Main article: orbifold

Something that looks locally like a manifold quotiented by the action of a finite group.OSp A Lie superalgebra.

18.19 Pp A momentumP 1. Parity, especially the parity symmetry.2. Short for periodic, a boundary condition on strings (as opposed to A for antiperiodic).3. Pseudoscalar (current)4. Momentum5. One of the bosonic elements of a supersymmetry algebra.p-brane A p+1 dimensional membrane, where p is a non-negative integer. The dimension of membranes is often

given by their space dimension, which is 1 less than their full spacetime dimension.PCAC partially conserved axial vector currentPDF Parton distribution function.PDG Particle Data Group.photino Main article: photino

A hypothetical supersymmetric partner of the photon.photon Main article: photon

The neutral spin 1 gauge boson of the electromagnetic field.PMNS Pontecorvo–Maki–Nakagawa–Sakata matrix for neutrino mixingPolyakov action Main article: Polyakov action

A modification of the Nambu–Goto action for strings that eliminates the square root.PQ Peccei–Quinn, as in Peccei–Quinn theory.pQCDPQCD Perturbative quantum chromodynamics.prepotential A function used to construct the vector superfield in supersymmetric gauge theory and Seiberg–Witten

theory.primary field A field killed by the positive weight operators of the Virasoro algebra (or similar algebra); in other

words, a lowest weight vector.Princeton string quartet DavidGross, JeffreyHarvey, EmilMartinec, andRyanRohm, who introduced the heterotic

string in 1985.PSL Projective special linear group.

18.20. Q 163

18.20 Qq A quark.

Q 1. The BRST operator.

2. A charge

3. One of the fermionic generators of a supersymmetry algebra.

quark Main article: quarkA strongly interacting elementary particle of spin 1/2.

QCD Main article: Quantum chromodynamics

QED Main article: Quantum electrodynamics

18.21 RR 1. Short for Ramond, as in Ramond sector.

2. A curvature tensorMain articles: Scalar curvature, Ricci curvature and Riemann curvature

3. An abbreviation for right (moving modes).

4. A radius

5. R-symmetry is a symmetry of extended supersymmetry algebras.

Ramond Named for Pierre Ramond.

Ramond algebra Main article: Ramond algebraA supersymmetric extension of the Virasoro algebra, similar to the Neveu–Schwarz algebra.

Rarita–Schwinger Main articles: Rarita–Schwinger equation and Rarita–Schwinger fieldRefers to spin 3/2 fermions.

Regge 1. Physicist Tullio Regge.

2. Regge trajectory: the squared mass of a hadronic resonance is roughly linear in the spin, with the constant ofproportionality called the Regge slope.

revolution Any new idea in string theory. In particular the first superstring revolution refers to the discoveries inthe mid 1980s such as the cancellation of gravitational anomalies and the heterotic string, and the secondsuperstring revolution refers to the discoveries in the mid 1990s, such as D-branes, M-theory, and matrixtheory and the AdS/CFT correspondence.

RG Renormalization group.

RGE Renormalization group equation.

RH Right-handed

R–NS A sector with Ramond conditions on left moving modes and Neveu–Schwarz conditions on right movingmodes.

RNS Ramond–Neveu–Schwarz, as in RNS formalism, a way of incorporating supersymmetry into string theory thatis supersymmetric on the world sheet.

R-parity Main article: R-parityA Z2 symmetry of supersymmetric models.

R-R Short for Ramond–Ramond sector

164 CHAPTER 18. GLOSSARY OF STRING THEORY

18.22 Ss 1. A strange quark.

2. A Mandelstam variable

S 1. An action

2. A scattering matrix.

3. The transformation τ → –1/τ of the upper half plane

4. Scalar (current)

5. Short for super or supersymmetric

S-brane A brane similar to a D-brane, with Dirichlet boundary conditions in the time direction.

S-duality Main article: S-dualityStrong–weak duality, a string duality relating theories with a large coupling constant to theories with a smallcoupling constant

SBB Standard Big Bang model of the universe

SCFT Superconformal field theory, a supersymmetric extension of conformal field theory

Schwarz Named for John Henry Schwarz

Seiberg duality Main article: Seiberg duality

SGA Abbreviation for Spectrum-generating algebra

short supermultiplet A supermultiplet (representation) related to BPS states

sigma model Main articles: Sigma model and Non-linear sigma modelA classical or quantum model based on the maps from a base manifold to a target manifold.

SL Special linear group

SLAC Stanford Linear Accelerator Center

SLC Stanford Linear Collider

slepton Main article: sleptonHypothetical supersymmetric partner of a lepton

SM Main article: Standard Model

sneutrino Main article: sneutrinoHypothetical supersymmetric partner of a neutrino

SO Special orthogonal group

Sp Symplectic group

sphaleron Main article: SphaleronStatic solution to the electroweak field equations

squark Main article: squarkSupersymmetric partner of a quark.

SSB Spontaneous symmetry breaking

SSM Standard solar model

stress–energy tensor Alternative name for the #energy–momentum tensor.

18.23. T 165

string field theory Main article: string field theory

SU Special unitary group

SUGRA Short for supergravity

superconformal algebra Main articles: Neveu–Schwarz algebra and Ramond algebraA supersymmetric analogue of the Virasoro algebra of conformal symmetries in 2 dimensions

superfield Main article: SuperfieldA supersymmetric analogue of a quantum or classical field

supergravity Main article: supergravityA supersymmetric extension of general relativity

supermultiplet Main article: SupermultipletA representation of a supersymmetry algebra

superpotential Main article: SuperpotentialA function of chiral superfield not depending on their superderivatives or spacetime derivatives, used to forma Lagrangian.

superspace Main article: superspaceA supersymmetric analogue of spacetime

superstring Main article: superstringA supersymmetric analogue of a string

supersymmetry Main articles: supersymmetry and supersymmetry algebraA generalization of a Lie superalgebra, where the Lie bracket [a,b] is sometimes given by ab+ba rather thanab–ba.

SUSY An abbreviation for supersymmetry.

SYM Supersymmetric Yang–Mills

18.23 Tt 1. A top quark.

2. A Mandelstam variable.

3. Time.

T 1. The energy–momentum tensor.

2. Time, especially the time symmetry.

3. The transformation τ → τ+1 of the upper half plane.

4. A torus.

5. The string tension.

6. Temperature.

7. Tensor (current)

T-duality Main article: T-dualityA string duality relating theories on a large spacetime to theories on a small spacetime. In particular it exchangestype IIA and IIB superstring theory.

tachyon Main article: TachyonA particle of imaginary mass moving faster than light.

166 CHAPTER 18. GLOSSARY OF STRING THEORY

ToE

TOE Theory of everything

type I

type II

type IIA

type IIB Main article: Type I string theoryMain article: Type II string theoryA type of superstring or the corresponding low-energy supergravity theory. The Roman numeral I or II refers tothe number of d=10 supersymmetries, and types IIA or IIB are distinguished by whether the supersymmetriesof left and right movers have opposite or identical chiralities.

18.24 U

u 1. An up quark.

2. A Mandelstam variable.

U A unitary group.

U-duality Main article: U-dualityShort for “unified duality”. A string duality relating two different string theories.

UED Universal extra dimensions

UV Short for ultra-violet, often referring to short-distance singularities.

18.25 V

V 1. A vertex operator.

2. Vector (current)

V-A Vector-Axial vector

vector superfield A type of superfield related to vector supermultiplets.

VEV Vacuum expectation value of an operator.

vielbein A frame

vierbein A frame in 4 dimensions. Sometimes used for a frame in an arbitrary number of dimension by authors whodo not care that “vier” means four in German.

Veneziano amplitude Main article: Veneziano amplitudeThe Euler beta function interpreted as a scattering amplitude.

vertex operator Main article: vertex operator

Virasoro algebra Main article: Virasoro algebraA central extension of the Witt algebra of polynomial vector fields on a circle.

18.26. W 167

18.26 Ww A complex number

W A W-boson

W-algebra Main article: W-algebraA sort of generalization of the Virasoro algebra

Weyl 1. Named after Hermann Weyl

2. A Weyl transformation is a rescaling of the world-sheet metric.

3. Weyl spinor, an element of a half-spin representation in even spacetime dimensions.

WIMP Weakly interacting massive particle

wino Main article: wino (particle)A hypothetical supersymmetric partner of the W-boson.

Witten Named for Edward Witten.

WMAP Wilkinson Microwave Anisotropy Probe

world sheet The 2-dimensional subset of spacetime swept out by a moving string.

world-volume The p+1-dimensional spacetime volume swept out by a p-brane, as in world-volume action.

WZNW

WZW Initials of Wess, Zumino, (Novikov), andWitten, as in the WZWmodel, a σ-model with a group as the targetspace.

18.27 XYZx A real number

X Used for coordinates in Minkowski space.

y A real number

YBE Yang–Baxter equation

YM Yang–Mills

z A complex number

Z 1. A partition function

2. The Z boson.

An element of the center of an extended supersymmetry algebra.

ZEUS Main article: ZEUS (particle detector)

zino Main article: gauginoA hypothetical supersymmetric partner of the Z-boson.

zweibein A frame in 2 dimensions

18.28 See also• List of string theory topics

168 CHAPTER 18. GLOSSARY OF STRING THEORY

18.29 References• Becker, Katrin, Becker, Melanie, and John H. Schwarz (2007) String Theory and M-Theory: A Modern Intro-duction . Cambridge University Press. ISBN 0-521-86069-5

• Binétruy, Pierre (2007) Supersymmetry: Theory, Experiment, and Cosmology. Oxford University Press. ISBN978-0-19-850954-7.

• Dine, Michael (2007) Supersymmetry and String Theory: Beyond the Standard Model. Cambridge UniversityPress. ISBN 0-521-85841-0.

• Paul H. Frampton (1974). Dual Resonance Models. Frontiers in Physics. ISBN 0-8053-2581-6.

• Michael Green, John H. Schwarz and Edward Witten (1987) Superstring theory. Cambridge University Press.The original textbook.

• Vol. 1: Introduction. ISBN 0-521-35752-7.• Vol. 2: Loop amplitudes, anomalies and phenomenology. ISBN 0-521-35753-5.

• Kiritsis, Elias (2007) String Theory in a Nutshell. Princeton University Press. ISBN 978-0-691-12230-4.

• Johnson, Clifford (2003). D-branes. Cambridge: Cambridge University Press. ISBN 0-521-80912-6.

• Polchinski, Joseph (1998) String Theory. Cambridge University Press.

• Vol. 1: An introduction to the bosonic string. ISBN 0-521-63303-6.• Vol. 2: Superstring theory and beyond. ISBN 0-521-63304-4.

• Szabo, Richard J. (Reprinted 2007) An Introduction to String Theory and D-brane Dynamics. Imperial CollegePress. ISBN 978-1-86094-427-7.

• Zwiebach, Barton (2004) A First Course in String Theory. Cambridge University Press. ISBN 0-521-83143-1.Contact author for errata.

18.30 External links• Particle physics glossary at interactions.org

Chapter 19

Glossary of systems theory

Main article: Systems theory

A glossary of terms as relating to systems theory.[1]

Contents :

• Top

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169

170 CHAPTER 19. GLOSSARY OF SYSTEMS THEORY

• U• V• W• X• Y• Z

19.1 A• Adaptive capacity: An important part of the resilience of systems in the face of a perturbation, helping tominimise loss of function in individual human, and collective social and biological systems.

• Allopoiesis is the process whereby a system produces something other than the system itself.

• Allostasis is the process of achieving stability, or homeostasis, through physiological or behavioral change.

• Autopoiesis is the process by which a system regenerates itself through the self-reproduction of its own ele-ments and of the network of interactions that characterize them. An autopoietic system renews, repairs, andreplicates or reproduces itself in a flow of matter and energy. Note: from a strictly Maturanian point of view,autopoiesis is an essential property of biological/living systems.

19.2 B• Black box is a technical term for a device or system or object when it is viewed primarily in terms of its inputand output characteristics.

• Boundaries: The parametric conditions, often vague, always subjectively stipulated, that delimit and define asystem and set it apart from its environment.

19.3 C• Cascading failure: failure in a system of interconnected parts, where the service provided depends on theoperation of a preceding part, and the failure of a preceding part can trigger the failure of successive parts.

• Closed system: a system which can exchange energy (as heat or work), but not matter, with its surroundings.

• Complexity: A systemic characteristic that stands for a large number of densely connected parts and multiplelevels of embeddedness and entanglement. Not to be confused with complicatedness, which denotes a situationor event that is not easy to understand, regardless of its degree of complexity.

• Culture: The result of individual learning processes that distinguish one social group of higher animals fromanother. In humans culture is the set of interrelated concepts, products and activities through which humansexpress themselves, interact with each other, and become aware of themselves and the world around them.

19.4 D• Development: The process of liberating a system from its previous set of limiting conditions. It is an amelio-ration of conditions or quality.

• Dissipative structure: A term invented by Ilya Prigogine to describe complex chemical structures undergoingthe process of chemical change through the dissipation of entropy into their environment, and the correspondingimportation of “negentropy” from their environment. Also known as syntropic systems.

19.5. E 171

19.5 E• Embeddedness: A state in which one system is nested in another system.

• Emergence: The appearance of novel characteristics exhibited on the level of the whole ensemble, but not bythe components in isolation.

• Enantiostasis is the ability of an open system, especially a living organism, to stabilize and conserve functionin spite of an unstable environment.

• Entanglement: A state in which the manner of being, or form of existence, of one system is inextricably tiedto that of another system or set of systems.

• Entropy: In physics, entropy is a measure of energy that is expended in a physical system but does no usefulwork, and tends to decrease the organizational order of the system.

• Environment: The context within which a system exists. It is composed of all things that are external to thesystem, and it includes everything that may affect the system, and may be affected by it at any given time.

• Evolution: A tendency toward greater structural complexity, ecological and/or organizational simplicity, moreefficient modes of operation, and greater dynamic harmony. A cosmic process specified by a fundamentaluniversal flow toward ever increasing complexity that manifests itself through particular events and sequencesof events that are not limited to the domain of biological phenomenon, but extend to include all aspects ofchange in open dynamic systems with a throughput of information and energy. In other words, evolutionrelates to the formation of stars from atoms, of Homo sapiens from the anthropoid apes, and the formation ofcomplex societies from rudimentary social systems.

• Evolutionary Systems: A form of systems design that responds to the need for a future-Design (ESD) creatingdesign praxis, that embraces not only human interests and life-spans, but those on planetary and evolutionaryplanes as well. The primary vehicle for the implementation of ESD is the Evolutionary Learning Community(ELC).

19.6 F• Feedback is a functional monitoring signal obtained from a given dynamic and continuous system. A feedbackfunction only makes sense if this monitoring signal is looped back into an eventual control structure within asystem and compared with a known desirable state. The difference between the feedback monitoring signaland the desirable state of the system gives the notion of error. The amount of error can guide corrective actionsin the system that can bring the system back to the desirable state.

19.7 H• Heterarchy: An ordering of things in which there is no single peak or leading element, and in which theelement that is dominant at a given time depends on the total situation. The term is often used in contrast tohierarchy, i.e. a vertical arrangement of entities (systems and their subsystems), usually ordered from the topdownwards rather than from the bottom upwards.

• Holarchy: A concept invented by Arthur Koestler to describe behavior that is partly a function of individualnature and partly a function of the nature of the embedding system, generally operating in a bottom upwardsfashion.

• Holism: A non-reductionist descriptive and investigative strategy for generating explanatory principles ofwhole systems. Attention is focused on the emergent properties of the whole rather than on the reductionistbehavior of the isolated parts. The approach typically involves and generates empathetic, experiential, andintuitive understanding, not merely analytic understanding, since by the definition of the approach, these formsare not truly separable (as nothing is).

172 CHAPTER 19. GLOSSARY OF SYSTEMS THEORY

• Holon (philosophy): A whole in itself as well as a part of a larger system.

• Homeorhesis is a concept encompassing dynamical systems which return to a trajectory, as opposed to systemswhich return to a particular state, which is termed homeostasis.

• Homeostasis is that property of either an open system or a closed system, especially a living organism, whichregulates its internal environment so as to maintain a stable, constant condition.

• Human Activity Systems: Designed social systems organized for a purpose, which they attain by carryingout specific functions.

19.8 I

• Isolated system: A system in which the total energy-mass is conserved without any external exchange hap-pening.

19.9 L

• Lowerarchy: A specific type of hierarchy involving a ‘bottom up’ arrangement of entities such that the feware influenced by the many.

19.10 M

• Metastability is the ability of a non-equilibrium state to persist for some period of time.

• Model building: A disciplined inquiry by which a conceptual (abstract) representation of a system is con-structed or a representation of expected outcomes/output is portrayed.

19.11 O

• Open system: A state and characteristics of that state in which a system continuously interacts with its envi-ronment. Open systems are those that maintain their state and exhibit the characteristics of openness previouslymentioned.

19.12 P

• Process is a naturally occurring or designed sequence of actions of an agent or changes of properties or at-tributes of an object or system.

• Process model: An organized arrangement of systems concepts and principles that portray the behavior of asystem through time. Its metaphor is the “motion-picture” of “movie” of the system.

19.13 R

• Reductionism: One kind of scientific orientation that seeks to understand phenomena by a) breaking themdown into their smallest possible parts: a process known as analytic reductionism, or conversely b) conflatingthem to a one-dimensional totality: a process known as holistic reductionism.

19.14. S 173

Open System Model (basics)

19.14 S• Self-organization is a process in which the internal organization of a system, normally an open system, in-creases in complexity without being guided or managed by an outside source.

• Self-organizing systems typically (though not always) display emergent properties.

• Steady state is a more general situation than Dynamic equilibrium. If a system is in steady state then therecently observed behaviour of the system will continue into the future. In stochastic systems, the probabilitiesthat various states will be repeated will remain constant.

• Strong emergence is a type of emergence in which the emergent property is irreducible to its individualconstituents.

• Subsystem: A major component of a system. It is made up of two or more interacting and interdependentcomponents. Subsystems of a system interact in order to attain their own purpose(s) and the purpose(s) of thesystem in which they are embedded.

• Suprasystem: The entity that is composed of a number of component systems organized in interacting rela-tionships in order to serve their embedding suprasystem.

• Sustainability: The ability of a system to maintain itself with no loss of function for extended periods of time.In human terms it is the creative and responsible stewardship of resources— human, Management natural, andfinancial — to generate stakeholder value while contributing to the well-being of current and future generationsof all beings.

174 CHAPTER 19. GLOSSARY OF SYSTEMS THEORY

• Synchrony: Also synchronicity. In engineering; concurrence of periods and/or phases; simultaneity of eventsor motions: contemporaneous occurrences. In evolutionary systems thinking; a fortunate coincidence of phe-nomenon and/or of events.

• Synergy: Also system. Synergy is the process by which a system generates emergent properties resulting inthe condition in which a system may be considered more than the sum of its parts, and equal to the sum of itsparts plus their relationships. This resulting condition can be said to be one of synergy.

• Syntony: In evolutionary systems thinking; evolutionary consonance; the occurrence and persistence of anevolutionarily tuned dynamic regime. Conscious intention aligned with evolutionary purpose; more loosely,the embodiment and manifestation of conscious evolution; a purposeful creative aligning and tuning with theevolutionary flows of one’s milieu. In traditional radio engineering; resonance.

• Syntropy The process of negentropy-importation. A syntropic system is a dissipative structure.

• Systems design A decision-oriented disciplined inquiry that aims at the construction of a model that is anabstract representation of a future system.

• Soft SystemsMethodology: Systemic approach for tackling real-world problematic situations, which providesa framework for users to deal with the kind of messy problem situations that lack a formal problem definition

19.15 W• Weak emergence is a type of emergence in which the emergent property is reducible to its individual con-stituents.

• White box is a technical term for a device or system analyzed or tested based on knowledge of its internalstructure (compare to Black box).

• Wholeness: In reference to systems, the condition in which systems are seen to be structurally divisible, butfunctionally indivisible wholes with emergent properties.

Contents :

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19.16. SEE ALSO 175

• L

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• S

• T

• U

• V

• W

• X

• Y

• Z

19.16 See also• Glossary of Unified Modeling Language terms

• List of basic science topics

• List of glossaries

• List of types of systems theory

19.17 References[1] Because systems language introduces many new terms essential to understanding how a system works, a glossary of many

of the significant terms is developed.

19.18 External links• Web Dictionary of Cybernetics and Systems from the Principia Cybernetica Web.

• The ASC Glossary of Cybernetics by the American Society for Cybernetics

• ASC Glossary on Cybernetics and Systems Theory by Stuart Umpleby (ed.) from the American Society forCybernetics.

• International Encyclopedia of Cybernetics and Systems, edited by Charles François, (1997) München: K. G.Saur.

Chapter 20

Glossary of tensor theory

This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:

• Tensor

• Tensor (intrinsic definition)

• Application of tensor theory in engineering science

For some history of the abstract theory see also Multilinear algebra.

20.1 Classical notationRicci calculus

The earliest foundation of tensor theory – tensor index notation.[1]

Tensor order

The components of a tensor with respect to a basis is an indexed array. The order of a tensor is the number of indicesneeded. Some texts may refer to the tensor order using the term degree or rank.

Rank

The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor. A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain thecorrect order.

Dyadic tensor

A dyadic tensor is a tensor of order two, and may be represented as a square matrix. In contrast, a dyad is specificallya dyadic tensor of rank one.

Einstein notation

This notation is based on the understanding that in a term in an expression contains a repeated index letter, the defaultinterpretation is that the product is summed over all permitted values of the index. For example if aij is a matrix,then under this convention aii is its trace. The Einstein convention is widely used in physics and engineering texts, tothe extent that if summation is not to be applied, it is normal to note that explicitly.

Kronecker delta

176

20.2. ALGEBRAIC NOTATION 177

Levi-Civita symbol

Covariant tensor

Contravariant tensor

The classical interpretation is by components. For example in the differential form aidxi the components ai are acovariant vector. That means all indices are lower; contravariant means all indices are upper.

Mixed tensor

This refers to any tensor that has both lower and upper indices.Cartesian tensorCartesian tensors are widely used in various branches of continuummechanics, such as fluid mechanics and elasticity.In classical continuum mechanics, the space of interest is usually 3-dimensional Euclidean space, as is the tangentspace at each point. If we restrict the local coordinates to be Cartesian coordinates with the same scale centered at thepoint of interest, the metric tensor is the Kronecker delta. This means that there is no need to distinguish covariant andcontravariant components, and furthermore there is no need to distinguish tensors and tensor densities. All Cartesian-tensor indices are written as subscripts. Cartesian tensors achieve considerable computational simplification at thecost of generality and of some theoretical insight.

Contraction of a tensor

Raising and lowering indices

Symmetric tensor

Antisymmetric tensor

Multiple cross products

20.2 Algebraic notation

This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol.

20.2.1 Tensor product

If v and w are vectors in vector spaces V andW respectively, then

v ⊗ w

is a tensor in

V ⊗W.

That is, the ⊗ operation is a binary operation, but it takes values into a fresh space (it is in a strong sense external).The ⊗ operation is a bilinear map; but no other conditions are applied to it.

20.2.2 Pure tensor

A pure tensor of V ⊗W is one that is of the form v ⊗ wIt could be written dyadically aibj, or more accurately aibj ei ⊗ fj, where the ei are a basis for V and the fj a basisforW. Therefore, unless V andW have the same dimension, the array of components need not be square. Such puretensors are not generic: if both V andW have dimension greater than 1, there will be tensors that are not pure, andthere will be non-linear conditions for a tensor to satisfy, to be pure. For more see Segre embedding.

178 CHAPTER 20. GLOSSARY OF TENSOR THEORY

20.2.3 Tensor algebra

In the tensor algebra T(V) of a vector space V, the operation

⊗

becomes a normal (internal) binary operation. A consequence is that T(V) has infinite dimension unless V hasdimension 0. The free algebra on a set X is for practical purposes the same as the tensor algebra on the vector spacewith X as basis.

20.2.4 Hodge star operator

20.2.5 Exterior power

The wedge product is the anti-symmetric form of the ⊗ operation. The quotient space of T(V) on which it becomesan internal operation is the exterior algebra of V ; it is a graded algebra, with the graded piece of weight k being calledthe k-th exterior power of V.

20.2.6 Symmetric power, symmetric algebra

This is the invariant way of constructing polynomial algebras.

20.3 Applications

Metric tensor

Strain tensor

Stress–energy tensor

20.4 Tensor field theory

Jacobian matrix

Tensor field

Tensor density

Lie derivative

Tensor derivative

Differential geometry

20.5 Abstract algebra

Tensor product of fields

This is an operation on fields, that does not always produce a field.

20.6. SPINORS 179

Tensor product of R-algebras

Clifford module

A representation of a Clifford algebra which gives a realisation of a Clifford algebra as a matrix algebra.

Tor functors

These are the derived functors of the tensor product, and feature strongly in homological algebra. The name comesfrom the torsion subgroup in abelian group theory.

Symbolic method of invariant theory

Derived category

Grothendieck’s six operations

These are highly abstract approaches used in some parts of geometry.

20.6 Spinors

See:

Spin group

Spin-c group

Spinor

Pin group

Pinors

Spinor field

Killing spinor

Spin manifold

20.7 References[1] Ricci, Gregorio; Levi-Civita, Tullio (March 1900), “Méthodes de calcul différentiel absolu et leurs applications” (PDF),

Mathematische Annalen (Springer) 54 (1–2): 125–201, doi:10.1007/BF01454201

20.8 Books• Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The MacmillanCompany, ISBN 0-486-64039-6

• Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus).ISBN 978-0-8133-4080-7.

180 CHAPTER 20. GLOSSARY OF TENSOR THEORY

• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers(Springer). ISBN 1-4020-1015-X.

• Lovelock, David; Hanno Rund (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover.ISBN 978-0-486-65840-7.

• Synge J.L., Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. ISBN 978-0-486-63612-2.

Chapter 21

Glossary of topology

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolutedistinction between different areas of topology, the focus here is on general topology. The following definitions arealso fundamental to algebraic topology, differential topology and geometric topology.See the article on topological spaces for basic definitions and examples, and see the article on topology for a briefhistory and description of the subject area. See Naive set theory, Axiomatic set theory, and Function for definitionsconcerning sets and functions. The following articles may also be useful. These either contain specialised vocabularywithin general topology or provide more detailed expositions of the definitions given below. The list of generaltopology topics and the list of examples in general topology will also be very helpful.

• Compact space

• Connected space

• Continuity

• Metric space

• Separated sets

• Separation axiom

• Topological space

• Uniform space

See also: Glossary of Riemannian and metric geometry

All spaces in this glossary are assumed to be topological spaces unless stated otherwise.Contents :

• Top

• 0–9

• A

• B

• C

• D

• E

• F

181

182 CHAPTER 21. GLOSSARY OF TOPOLOGY

• G

• H

• I

• J

• K

• L

• M

• N

• O

• P

• Q

• R

• S

• T

• U

• V

• W

• X

• Y

• Z

21.1 A

Absolutely closed See H-closed

Accessible See T1 .

Accumulation point See limit point.

Alexandrov topology The topology of a space X is an Alexandrov topology (or is finitely generated) if arbitraryintersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, againequivalently, if the open sets are the upper sets of a poset.[1]

Almost discrete A space is almost discrete if every open set is closed (hence clopen). The almost discrete spacesare precisely the finitely generated zero-dimensional spaces.

Approach space An approach space is a generalization of metric space based on point-to-set distances, instead ofpoint-to-point.

21.2. B 183

21.2 BBaire space This has two distinct common meanings:

1. A space is a Baire space if the intersection of any countable collection of dense open sets is dense; seeBaire space.

2. Baire space is the set of all functions from the natural numbers to the natural numbers, with the topologyof pointwise convergence; see Baire space (set theory).

Base A collection B of open sets is a base (or basis) for a topology τ if every open set in τ is a union of sets in B .The topology τ is the smallest topology on X containing B and is said to be generated by B .

Basis See Base.

Borel algebra The Borel algebra on a topological space (X, τ) is the smallest σ -algebra containing all the open sets.It is obtained by taking intersection of all σ -algebras on X containing τ .

Borel set A Borel set is an element of a Borel algebra.

Boundary The boundary (or frontier) of a set is the set’s closure minus its interior. Equivalently, the boundary of aset is the intersection of its closure with the closure of its complement. Boundary of a set A is denoted by ∂Aor bd A .

Bounded A set in a metric space is bounded if it has finite diameter. Equivalently, a set is bounded if it is containedin some open ball of finite radius. A function taking values in a metric space is bounded if its image is abounded set.

21.3 CCategory of topological spaces The categoryTop has topological spaces as objects and continuousmaps asmorphisms.

Cauchy sequence A sequence {xn} in a metric space (M, d) is a Cauchy sequence if, for every positive real numberr, there is an integer N such that for all integers m, n > N, we have d(xm, xn) < r.

Clopen set A set is clopen if it is both open and closed.

Closed ball If (M, d) is a metric space, a closed ball is a set of the form D(x; r) := {y inM : d(x, y) ≤ r}, where x isinM and r is a positive real number, the radius of the ball. A closed ball of radius r is a closed r-ball. Everyclosed ball is a closed set in the topology induced on M by d. Note that the closed ball D(x; r) might not beequal to the closure of the open ball B(x; r).

Closed set A set is closed if its complement is a member of the topology.

Closed function A function from one space to another is closed if the image of every closed set is closed.

Closure The closure of a set is the smallest closed set containing the original set. It is equal to the intersection of allclosed sets which contain it. An element of the closure of a set S is a point of closure of S.

Closure operator See Kuratowski closure axioms.

Coarser topology If X is a set, and if T1 and T2 are topologies on X, then T1 is coarser (or smaller, weaker) thanT2 if T1 is contained in T2. Beware, some authors, especially analysts, use the term stronger.

Comeagre A subset A of a spaceX is comeagre (comeager) if its complementX\A is meagre. Also called residual.

184 CHAPTER 21. GLOSSARY OF TOPOLOGY

Compact A space is compact if every open cover has a finite subcover. Every compact space is Lindelöf andparacompact. Therefore, every compact Hausdorff space is normal. See also quasicompact.

Compact-open topology The compact-open topology on the set C(X, Y) of all continuous maps between two spacesX and Y is defined as follows: given a compact subset K of X and an open subset U of Y, let V(K, U) denotethe set of all maps f in C(X, Y) such that f(K) is contained in U. Then the collection of all such V(K, U) is asubbase for the compact-open topology.

Complete A metric space is complete if every Cauchy sequence converges.

Completely metrizable/completely metrisable See complete space.

Completely normal A space is completely normal if any two separated sets have disjoint neighbourhoods.

Completely normal Hausdorff A completely normal Hausdorff space (or T5 space) is a completely normal T1

space. (A completely normal space is Hausdorff if and only if it is T1, so the terminology is consistent.) Everycompletely normal Hausdorff space is normal Hausdorff.

Completely regular A space is completely regular if, whenever C is a closed set and x is a point not in C, then Cand {x} are functionally separated.

Completely T3 See Tychonoff.

Component See Connected component/Path-connected component.

Connected A space is connected if it is not the union of a pair of disjoint nonempty open sets. Equivalently, a spaceis connected if the only clopen sets are the whole space and the empty set.

Connected component A connected component of a space is a maximal nonempty connected subspace. Each con-nected component is closed, and the set of connected components of a space is a partition of that space.

Continuous A function from one space to another is continuous if the preimage of every open set is open.

Continuum A space is called a continuum if it a compact, connected Hausdorff space.

Contractible A space X is contractible if the identity map on X is homotopic to a constant map. Every contractiblespace is simply connected.

Coproduct topology If {Xi} is a collection of spaces and X is the (set-theoretic) disjoint union of {Xi}, then thecoproduct topology (or disjoint union topology, topological sum of the Xi) on X is the finest topology forwhich all the injection maps are continuous.

Cosmic space A continuous image of some separable metric space.[2]

Countable chain condition A space X satisfies the countable chain condition if every family of non-empty, pairs-wise disjoint open sets is countable.

Countably compact A space is countably compact if every countable open cover has a finite subcover. Every count-ably compact space is pseudocompact and weakly countably compact.

Countably locally finite A collection of subsets of a space X is countably locally finite (or σ-locally finite) if it isthe union of a countable collection of locally finite collections of subsets of X.

Cover A collection of subsets of a space is a cover (or covering) of that space if the union of the collection is thewhole space.

Covering See Cover.

Cut point If X is a connected space with more than one point, then a point x of X is a cut point if the subspace X −{x} is disconnected.

21.4. D 185

21.4 D

Dense set A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is denseif its closure is the whole space.

Dense-in-itself set A set is dense-in-itself if it has no isolated point.

Density the minimal cardinality of a dense subset of a topological space. A set of density ℵ0 is a separable space.[3]

Derived set If X is a space and S is a subset of X, the derived set of S in X is the set of limit points of S in X.

Developable space A topological space with a development.[4]

Development A countable collection of open covers of a topological space, such that for any closed set C and anypoint p in its complement there exists a cover in the collection such that every neighbourhood of p in the coveris disjoint from C.[4]

Diameter If (M, d) is a metric space and S is a subset ofM, the diameter of S is the supremum of the distances d(x,y), where x and y range over S.

Discrete metric The discrete metric on a set X is the function d : X × X→ R such that for all x, y in X, d(x, x) = 0and d(x, y) = 1 if x ≠ y. The discrete metric induces the discrete topology on X.

Discrete space A space X is discrete if every subset of X is open. We say that X carries the discrete topology.[5]

Discrete topology See discrete space.

Disjoint union topology See Coproduct topology.

Dispersion point If X is a connected space with more than one point, then a point x of X is a dispersion point if thesubspace X − {x} is hereditarily disconnected (its only connected components are the one-point sets).

Distance See metric space.

Dunce hat (topology)

21.5 E

Entourage See Uniform space.

Exterior The exterior of a set is the interior of its complement.

21.6 F

Fσ set An Fσ set is a countable union of closed sets.[6]

Filter A filter on a space X is a nonempty family F of subsets of X such that the following conditions hold:

1. The empty set is not in F.2. The intersection of any finite number of elements of F is again in F.3. If A is in F and if B contains A, then B is in F.

186 CHAPTER 21. GLOSSARY OF TOPOLOGY

Final topology On a set X with respect to a family of functions into X , is the finest topology on X which makesthose functions continuous.[7]

Fine topology (potential theory) On Euclidean spaceRn , the coarsest topology making all subharmonic functions(equivalently all superharmonic functions) continuous.[8]

Finer topology If X is a set, and if T1 and T2 are topologies on X, then T2 is finer (or larger, stronger) than T1 ifT2 contains T1. Beware, some authors, especially analysts, use the term weaker.

Finitely generated See Alexandrov topology.

First category SeeMeagre.

First-countable A space is first-countable if every point has a countable local base.

Fréchet See T1.

Frontier See Boundary.

Full set A compact subset K of the complex plane is called full if its complement is connected. For example, theclosed unit disk is full, while the unit circle is not.

Functionally separated Two sets A and B in a space X are functionally separated if there is a continuous map f: X→ [0, 1] such that f(A) = 0 and f(B) = 1.

21.7 G

Gδ set A Gδ set or inner limiting set is a countable intersection of open sets.[6]

Gδ space A space in which every closed set is a Gδ set.[6]

Generic point A generic point for a closed set is a point for which the closed set is the closure of the singleton setcontaining that point.[9]

21.8 H

Hausdorff A Hausdorff space (or T2 space) is one in which every two distinct points have disjoint neighbourhoods.Every Hausdorff space is T1.

H-closed A space is H-closed, or Hausdorff closed or absolutely closed, if it is closed in every Hausdorff spacecontaining it.

Hereditarily P A space is hereditarily P for some property P if every subspace is also P.

Hereditary A property of spaces is said to be hereditary if whenever a space has that property, then so does everysubspace of it.[10] For example, second-countability is a hereditary property.

Homeomorphism If X and Y are spaces, a homeomorphism from X to Y is a bijective function f : X → Y suchthat f and f−1 are continuous. The spaces X and Y are then said to be homeomorphic. From the standpointof topology, homeomorphic spaces are identical.

Homogeneous A space X is homogeneous if, for every x and y in X, there is a homeomorphism f : X→ X such thatf(x) = y. Intuitively, the space looks the same at every point. Every topological group is homogeneous.

21.9. I 187

Homotopic maps Two continuous maps f, g : X→ Y are homotopic (in Y) if there is a continuous map H : X × [0,1] → Y such that H(x, 0) = f(x) and H(x, 1) = g(x) for all x in X. Here, X × [0, 1] is given the product topology.The function H is called a homotopy (in Y) between f and g.

Homotopy See Homotopic maps.

Hyper-connected A space is hyper-connected if no two non-empty open sets are disjoint[11] Every hyper-connectedspace is connected.[11]

21.9 I

Identification map See Quotient map.

Identification space See Quotient space.

Indiscrete space See Trivial topology.

Infinite-dimensional topology See Hilbert manifold and Q-manifolds, i.e. (generalized) manifolds modelled onthe Hilbert space and on the Hilbert cube respectively.

Inner limiting set A Gδ set.[6]

Interior The interior of a set is the largest open set contained in the original set. It is equal to the union of all opensets contained in it. An element of the interior of a set S is an interior point of S.

Interior point See Interior.

Isolated point A point x is an isolated point if the singleton {x} is open. More generally, if S is a subset of a spaceX, and if x is a point of S, then x is an isolated point of S if {x} is open in the subspace topology on S.

Isometric isomorphism If M1 and M2 are metric spaces, an isometric isomorphism from M1 to M2 is a bijectiveisometry f : M1 →M2. The metric spaces are then said to be isometrically isomorphic. From the standpointof metric space theory, isometrically isomorphic spaces are identical.

Isometry If (M1, d1) and (M2, d2) are metric spaces, an isometry from M1 to M2 is a function f : M1 → M2 suchthat d2(f(x), f(y)) = d1(x, y) for all x, y in M1. Every isometry is injective, although not every isometry issurjective.

21.10 K

Kolmogorov axiom See T0.

Kuratowski closure axioms The Kuratowski closure axioms is a set of axioms satisfied by the function which takeseach subset of X to its closure:

1. Isotonicity: Every set is contained in its closure.2. Idempotence: The closure of the closure of a set is equal to the closure of that set.3. Preservation of binary unions: The closure of the union of two sets is the union of their closures.4. Preservation of nullary unions: The closure of the empty set is empty.

If c is a function from the power set of X to itself, then c is a closure operator if it satisfies the Kuratowski closureaxioms. The Kuratowski closure axioms can then be used to define a topology on X by declaring the closedsets to be the fixed points of this operator, i.e. a set A is closed if and only if c(A) = A.

Kolmogorov topology TKol = {R,∅ }∪{(a,∞): a is real number}; the pair (R,TKol) is named Kolmogorov Straight.

188 CHAPTER 21. GLOSSARY OF TOPOLOGY

21.11 LL-space An L-space is a hereditarily Lindelöf space which is not hereditarily separable. A Suslin line would be an

L-space.[12]

Larger topology See Finer topology.

Limit point A point x in a space X is a limit point of a subset S if every open set containing x also contains a pointof S other than x itself. This is equivalent to requiring that every neighbourhood of x contains a point of S otherthan x itself.

Limit point compact SeeWeakly countably compact.

Lindelöf A space is Lindelöf if every open cover has a countable subcover.

Local base A set B of neighbourhoods of a point x of a space X is a local base (or local basis, neighbourhoodbase, neighbourhood basis) at x if every neighbourhood of x contains some member of B.

Local basis See Local base.

Locally (P) space There are two definitions for a space to be “locally (P)" where (P) is a topological or set-theoreticproperty: that each point has a neighbourhood with property (P), or that every point has a neighourbood basefor which each member has property (P). The first definition is usually taken for locally compact, countablycompact, metrisable, separable, countable; the second for locally connected.[13]

Locally closed subset A subset of a topological space that is the intersection of an open and a closed subset. Equiv-alently, it is a relatively open subset of its closure.

Locally compact A space is locally compact if every point has a compact neighbourhood: the alternative definitionthat each point has a local base consisting of compact neighbourhoods is sometimes used: these are equivalentfor Hausdorff spaces.[13] Every locally compact Hausdorff space is Tychonoff.

Locally connected A space is locally connected if every point has a local base consisting of connected neighbourhoods.[13]

Locally finite A collection of subsets of a space is locally finite if every point has a neighbourhood which hasnonempty intersection with only finitely many of the subsets. See also countably locally finite, point finite.

Locally metrizable/Locally metrisable A space is locallymetrizable if every point has ametrizable neighbourhood.[13]

Locally path-connected A space is locally path-connected if every point has a local base consisting of path-connectedneighbourhoods.[13] A locally path-connected space is connected if and only if it is path-connected.

Locally simply connected A space is locally simply connected if every point has a local base consisting of simplyconnected neighbourhoods.

Loop If x is a point in a space X, a loop at x in X (or a loop in X with basepoint x) is a path f in X, such that f(0) =f(1) = x. Equivalently, a loop in X is a continuous map from the unit circle S1 into X.

21.12 MMeagre If X is a space and A is a subset of X, then A is meagre in X (or of first category in X) if it is the countable

union of nowhere dense sets. If A is not meagre in X, A is of second category in X.[14]

Metacompact A space is metacompact if every open cover has a point finite open refinement.

21.13. N 189

Metric SeeMetric space.

Metric invariant A metric invariant is a property which is preserved under isometric isomorphism.

Metric map If X and Y are metric spaces with metrics dX and dY respectively, then a metric map is a function ffrom X to Y, such that for any points x and y in X, dY(f(x), f(y)) ≤ dX(x, y). A metric map is strictly metricif the above inequality is strict for all x and y in X.

Metric space A metric space (M, d) is a set M equipped with a function d : M × M → R satisfying the followingaxioms for all x, y, and z in M:

1. d(x, y) ≥ 02. d(x, x) = 03. if d(x, y) = 0 then x = y (identity of indiscernibles)4. d(x, y) = d(y, x) (symmetry)5. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

The function d is ametric onM, and d(x, y) is the distance between x and y. The collection of all openballs of M is a base for a topology on M; this is the topology on M induced by d. Every metric space isHausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable.

Metrizable/Metrisable A space is metrizable if it is homeomorphic to a metric space. Every metrizable space isHausdorff and paracompact (and hence normal and Tychonoff). Every metrizable space is first-countable.

Monolith Every non-empty ultra-connected compact space X has a largest proper open subset; this subset is calledamonolith.

Moore space A Moore space is a developable regular Hausdorff space.[4]

21.13 NNeighbourhood/Neighborhood A neighbourhood of a point x is a set containing an open set which in turn contains

the point x. More generally, a neighbourhood of a set S is a set containing an open set which in turn containsthe set S. A neighbourhood of a point x is thus a neighbourhood of the singleton set {x}. (Note that under thisdefinition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; becareful to note conventions.)

Neighbourhood base/basis See Local base.

Neighbourhood system for a point x A neighbourhood system at a point x in a space is the collection of all neigh-bourhoods of x.

Net A net in a space X is a map from a directed set A to X. A net from A to X is usually denoted (xα), where α is anindex variable ranging over A. Every sequence is a net, taking A to be the directed set of natural numbers withthe usual ordering.

Normal A space is normal if any two disjoint closed sets have disjoint neighbourhoods.[6] Every normal space admitsa partition of unity.

Normal Hausdorff A normal Hausdorff space (or T4 space) is a normal T1 space. (A normal space is Hausdorff ifand only if it is T1, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff.

Nowhere dense A nowhere dense set is a set whose closure has empty interior.

190 CHAPTER 21. GLOSSARY OF TOPOLOGY

21.14 OOpen cover An open cover is a cover consisting of open sets.[4]

Open ball If (M, d) is a metric space, an open ball is a set of the form B(x; r) := {y in M : d(x, y) < r}, where x isin M and r is a positive real number, the radius of the ball. An open ball of radius r is an open r-ball. Everyopen ball is an open set in the topology on M induced by d.

Open condition See open property.

Open set An open set is a member of the topology.

Open function A function from one space to another is open if the image of every open set is open.

Open property A property of points in a topological space is said to be “open” if those points which possess it forman open set. Such conditions often take a common form, and that form can be said to be an open condition;for example, in metric spaces, one defines an open ball as above, and says that “strict inequality is an opencondition”.

21.15 PParacompact A space is paracompact if every open cover has a locally finite open refinement. Paracompact implies

metacompact.[15] Paracompact Hausdorff spaces are normal.[16]

Partition of unity A partition of unity of a space X is a set of continuous functions from X to [0, 1] such that anypoint has a neighbourhood where all but a finite number of the functions are identically zero, and the sum ofall the functions on the entire space is identically 1.

Path A path in a space X is a continuous map f from the closed unit interval [0, 1] into X. The point f(0) is theinitial point of f; the point f(1) is the terminal point of f.[11]

Path-connected A space X is path-connected if, for every two points x, y in X, there is a path f from x to y, i.e., apath with initial point f(0) = x and terminal point f(1) = y. Every path-connected space is connected.[11]

Path-connected component A path-connected component of a space is a maximal nonempty path-connected sub-space. The set of path-connected components of a space is a partition of that space, which is finer than thepartition into connected components.[11] The set of path-connected components of a space X is denoted π0(X).

Perfectly normal a normal space which is also a Gδ.[6]

π-base A collection B of nonempty open sets is a π-base for a topology τ if every nonempty open set in τ includesa set from B.[17]

Point A point is an element of a topological space. More generally, a point is an element of any set with an underlyingtopological structure; e.g. an element of a metric space or a topological group is also a “point”.

Point of closure See Closure.

Polish A space is Polish if it is separable and completely metrizable, i.e. if it is homeomorphic to a separable andcomplete metric space.

Polyadic A space is polyadic if it is the continuous image of the power of a one-point compactification of a locallycompact, non-compact Hausdorff space.

P-point A point of a topological space is a P-point if its filter of neighbourhoods is closed under countable intersec-tions.

21.16. Q 191

Pre-compact See Relatively compact.

Prodiscrete topology The prodiscrete topology on a product AG is the product topology when each factor A is giventhe discrete topology.[18]

Product topology If {Xi} is a collection of spaces and X is the (set-theoretic) product of {Xi}, then the producttopology on X is the coarsest topology for which all the projection maps are continuous.

Proper function/mapping A continuous function f from a space X to a space Y is proper if f−1(C) is a compactset in X for any compact subspace C of Y.

Proximity space Aproximity space (X, δ) is a setX equippedwith a binary relation δ between subsets ofX satisfyingthe following properties:

For all subsets A, B and C of X,

1. A δ B implies B δ A2. A δ B implies A is non-empty3. If A and B have non-empty intersection, then A δ B4. A δ (B ∪ C) iff (A δ B or A δ C)5. If, for all subsets E of X, we have (A δ E or B δ E), then we must have A δ (X − B)

Pseudocompact A space is pseudocompact if every real-valued continuous function on the space is bounded.

Pseudometric See Pseudometric space.

Pseudometric space A pseudometric space (M, d) is a setM equipped with a function d : M ×M →R satisfying allthe conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometricspace may be “infinitely close” without being identical. The function d is a pseudometric onM. Every metricis a pseudometric.

Punctured neighbourhood/Punctured neighborhood Apunctured neighbourhood of a point x is a neighbourhoodof x, minus {x}. For instance, the interval (−1, 1) = {y : −1 < y < 1} is a neighbourhood of x = 0 in the realline, so the set (−1, 0) ∪ (0, 1) = (−1, 1) − {0} is a punctured neighbourhood of 0.

21.16 Q

Quasicompact See compact. Some authors define “compact” to include the Hausdorff separation axiom, and theyuse the term quasicompact to mean what we call in this glossary simply “compact” (without the Hausdorffaxiom). This convention is most commonly found in French, and branches of mathematics heavily influencedby the French.

Quotient map If X and Y are spaces, and if f is a surjection from X to Y, then f is a quotient map (or identificationmap) if, for every subset U of Y, U is open in Y if and only if f −1(U) is open in X. In other words, Y hasthe f-strong topology. Equivalently, f is a quotient map if and only if it is the transfinite composition of mapsX → X/Z , where Z ⊂ X is a subset. Note that this doesn't imply that f is an open function.

Quotient space If X is a space, Y is a set, and f : X→ Y is any surjective function, then the quotient topology on Yinduced by f is the finest topology for which f is continuous. The space X is a quotient space or identificationspace. By definition, f is a quotient map. The most common example of this is to consider an equivalencerelation on X, with Y the set of equivalence classes and f the natural projection map. This construction is dualto the construction of the subspace topology.

192 CHAPTER 21. GLOSSARY OF TOPOLOGY

21.17 RRefinement A cover K is a refinement of a cover L if every member of K is a subset of some member of L.

Regular A space is regular if, whenever C is a closed set and x is a point not in C, then C and x have disjointneighbourhoods.

Regular Hausdorff A space is regular Hausdorff (or T3) if it is a regular T0 space. (A regular space is Hausdorffif and only if it is T0, so the terminology is consistent.)

Regular open A subset of a space X is regular open if it equals the interior of its closure; dually, a regular closed setis equal to the closure of its interior.[19] An example of a non-regular open set is the set U = (0,1) ∪ (1,2) in Rwith its normal topology, since 1 is in the interior of the closure of U, but not in U. The regular open subsetsof a space form a complete Boolean algebra.[19]

Relatively compact A subset Y of a space X is relatively compact in X if the closure of Y in X is compact.

Residual If X is a space and A is a subset of X, then A is residual in X if the complement of A is meagre in X. Alsocalled comeagre or comeager.

Resolvable A topological space is called resolvable if it is expressible as the union of two disjoint dense subsets.

Rim-compact A space is rim-compact if it has a base of open sets whose boundaries are compact.

21.18 SS-space An S-space is a hereditarily separable space which is not hereditarily Lindelöf.[12]

Scattered A space X is scattered if every nonempty subset A of X contains a point isolated in A.

Scott The Scott topology on a poset is that in which the open sets are those Upper sets inaccessible by directedjoins.[20]

Second category SeeMeagre.

Second-countable A space is second-countable or perfectly separable if it has a countable base for its topology.[6]Every second-countable space is first-countable, separable, and Lindelöf.

Semilocally simply connected A space X is semilocally simply connected if, for every point x in X, there is aneighbourhood U of x such that every loop at x in U is homotopic in X to the constant loop x. Every simplyconnected space and every locally simply connected space is semilocally simply connected. (Compare withlocally simply connected; here, the homotopy is allowed to live in X, whereas in the definition of locally simplyconnected, the homotopy must live in U.)

Semiregular A space is semiregular if the regular open sets form a base.

Separable A space is separable if it has a countable dense subset.[6][14]

Separated Two sets A and B are separated if each is disjoint from the other’s closure.

Sequentially compact A space is sequentially compact if every sequence has a convergent subsequence. Everysequentially compact space is countably compact, and every first-countable, countably compact space is se-quentially compact.

Short map See metric map

Simply connected A space is simply connected if it is path-connected and every loop is homotopic to a constantmap.

21.19. T 193

Smaller topology See Coarser topology.

Sober In a sober space, every irreducible closed subset is the closure of exactly one point: that is, has a uniquegeneric point.[21]

Star The star of a point in a given cover of a topological space is the union of all the sets in the cover that containthe point. See star refinement.

f -Strong topologyLet f : X → Y be a map of topological spaces. We say that Y has the f -strong topology if, for every subsetU ⊂ Y , one has that U is open in Y if and only if f−1(U) is open in X

Stronger topology See Finer topology. Beware, some authors, especially analysts, use the term weaker topology.

Subbase A collection of open sets is a subbase (or subbasis) for a topology if every non-empty proper open set inthe topology is a union of finite intersections of sets in the subbase. If B is any collection of subsets of a setX, the topology on X generated by B is the smallest topology containing B; this topology consists of the emptyset, X and all unions of finite intersections of elements of B.

Subbasis See Subbase.

Subcover A cover K is a subcover (or subcovering) of a cover L if every member of K is a member of L.

Subcovering See Subcover.

Submaximal space A topological space is said to be submaximal if every subset of it is locally closed, that is, everysubset is the intersection of an open set and a closed set.

Here are some facts about submaximality as a property of topological spaces:

• Every door space is submaximal.

• Every submaximal space is weakly submaximal viz every finite set is locally closed.

• Every submaximal space is irresolvable[22]

Subspace If T is a topology on a space X, and if A is a subset of X, then the subspace topology on A induced byT consists of all intersections of open sets in T with A. This construction is dual to the construction of thequotient topology.

21.19 T

T0 A space is T0 (orKolmogorov) if for every pair of distinct points x and y in the space, either there is an open setcontaining x but not y, or there is an open set containing y but not x.

T1 A space is T1 (or Fréchet or accessible) if for every pair of distinct points x and y in the space, there is an openset containing x but not y. (Compare with T0; here, we are allowed to specify which point will be contained inthe open set.) Equivalently, a space is T1 if all its singletons are closed. Every T1 space is T0.

T2 See Hausdorff space.

T3 See Regular Hausdorff.

T₃½ See Tychonoff space.

194 CHAPTER 21. GLOSSARY OF TOPOLOGY

T4 See Normal Hausdorff.

T5 See Completely normal Hausdorff.

Top See Category of topological spaces.

Topological invariant A topological invariant is a property which is preserved under homeomorphism. For exam-ple, compactness and connectedness are topological properties, whereas boundedness and completeness arenot. Algebraic topology is the study of topologically invariant abstract algebra constructions on topologicalspaces.

Topological space A topological space (X, T) is a set X equipped with a collection T of subsets of X satisfying thefollowing axioms:

1. The empty set and X are in T.2. The union of any collection of sets in T is also in T.3. The intersection of any pair of sets in T is also in T.

The collection T is a topology on X.

Topological sum See Coproduct topology.

Topologically complete Completely metrizable spaces (i. e. topological spaces homeomorphic to complete metricspaces) are often called topologically complete; sometimes the term is also used for Čech-complete spaces orcompletely uniformizable spaces.

Topology See Topological space.

Totally bounded A metric space M is totally bounded if, for every r > 0, there exist a finite cover of M by openballs of radius r. A metric space is compact if and only if it is complete and totally bounded.

Totally disconnected A space is totally disconnected if it has no connected subset with more than one point.

Trivial topology The trivial topology (or indiscrete topology) on a set X consists of precisely the empty set and theentire space X.

Tychonoff A Tychonoff space (or completely regular Hausdorff space, completely T3 space, T₃.₅ space) is acompletely regular T0 space. (A completely regular space is Hausdorff if and only if it is T0, so the terminologyis consistent.) Every Tychonoff space is regular Hausdorff.

21.20 UUltra-connected A space is ultra-connected if no two non-empty closed sets are disjoint.[11] Every ultra-connected

space is path-connected.

Ultrametric A metric is an ultrametric if it satisfies the following stronger version of the triangle inequality: for allx, y, z in M, d(x, z) ≤ max(d(x, y), d(y, z)).

Uniform isomorphism If X and Y are uniform spaces, a uniform isomorphism from X to Y is a bijective functionf : X→ Y such that f and f−1 are uniformly continuous. The spaces are then said to be uniformly isomorphicand share the same uniform properties.

Uniformizable/Uniformisable A space is uniformizable if it is homeomorphic to a uniform space.

21.21. W 195

Uniform space A uniform space is a set U equipped with a nonempty collection Φ of subsets of the Cartesianproduct X × X satisfying the following axioms:

1. if U is in Φ, then U contains { (x, x) | x in X }.2. if U is in Φ, then { (y, x) | (x, y) in U } is also in Φ3. if U is in Φ and V is a subset of X × X which contains U, then V is in Φ4. if U and V are in Φ, then U ∩ V is in Φ5. if U is in Φ, then there exists V in Φ such that, whenever (x, y) and (y, z) are in V, then (x, z) is in

U.

The elements of Φ are called entourages, and Φ itself is called a uniform structure on U.

Uniform structure See Uniform space.

21.21 WWeak topology The weak topology on a set, with respect to a collection of functions from that set into topological

spaces, is the coarsest topology on the set which makes all the functions continuous.

Weaker topology See Coarser topology. Beware, some authors, especially analysts, use the term stronger topol-ogy.

Weakly countably compact A space is weakly countably compact (or limit point compact) if every infinite subsethas a limit point.

Weakly hereditary A property of spaces is said to be weakly hereditary if whenever a space has that property, thenso does every closed subspace of it. For example, compactness and the Lindelöf property are both weaklyhereditary properties, although neither is hereditary.

Weight The weight of a space X is the smallest cardinal number κ such that X has a base of cardinal κ. (Note thatsuch a cardinal number exists, because the entire topology forms a base, and because the class of cardinalnumbers is well-ordered.)

Well-connected See Ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)

21.22 ZZero-dimensional A space is zero-dimensional if it has a base of clopen sets.[23]

21.23 References[1] Vickers (1989) p.22

[2] Deza, Michel Marie; Deza, Elena (2012). Encyclopedia of Distances. Springer-Verlag. p. 64. ISBN 3642309585.

[3] Nagata (1985) p.104

[4] Steen & Seebach (1978) p.163

[5] Steen & Seebach (1978) p.41

[6] Steen & Seebach (1978) p.162

[7] Willard, Stephen (1970). General Topology. Addison-Wesley Series in Mathematics. Reading, MA: Addison-Wesley. Zbl0205.26601.

[8] Conway, John B. (1995). Functions of One Complex Variable II. Graduate Texts in Mathematics 159. Springer-Verlag. pp.367–376. ISBN 0-387-94460-5. Zbl 0887.30003.

196 CHAPTER 21. GLOSSARY OF TOPOLOGY

[9] Vickers (1989) p.65

[10] Steen & Seebach p.4

[11] Steen & Seebach (1978) p.29

[12] Gabbay, Dov M.; Kanamori, Akihiro; Woods, John Hayden, eds. (2012). Sets and Extensions in the Twentieth Century.Elsevier. p. 290. ISBN 0444516212.

[13] Hart et al (2004) p.65

[14] Steen & Seebach (1978) p.7

[15] Steen & Seebach (1978) p.23

[16] Steen & Seebach (1978) p.25

[17] Hart, Nagata, Vaughan Sect. d-22, page 227

[18] Ceccherini-Silberstein, Tullio; Coornaert, Michel (2010). Cellular automata and groups. Springer Monographs in Mathe-matics. Berlin: Springer-Verlag. p. 3. ISBN 978-3-642-14033-4. Zbl 1218.37004.

[19] Steen & Seebach (1978) p.6

[20] Vickers (1989) p.95

[21] Vickers (1989) p.66

[22] Miroslav Hušek; J. van Mill (2002), Recent progress in general topology, Recent Progress in General Topology 2, Elsevier,p. 21, ISBN 0-444-50980-1

[23] Steen & Seebach (1978) p.33

• Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004). Encyclopedia of general topology. Elsevier.ISBN 978-0-444-50355-8.

• Kunen, Kenneth; Vaughan, Jerry E. (editors). Handbook of Set-Theoretic Topology. North-Holland. ISBN0-444-86580-2.

• Nagata, Jun-iti (1985). Modern general topology. North-Holland Mathematical Library 33 (2nd revised ed.).Amsterdam-New York-Oxford: North-Holland. ISBN 0080933793. Zbl 0598.54001.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (Dover reprint of 1978 ed.).Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446.

• Vickers, Steven (1989). Topology via Logic. Cambridge Tracts in Theoretic Computer Science 5. ISBN0-521-36062-5. Zbl 0668.54001.

• Willard, Stephen (1970). General Topology. Addison-Wesley Series in Mathematics. Reading, MA: Addison-Wesley. ISBN 978-0-201-08707-9. Zbl 0205.26601. Also available as Dover reprint.

21.24 External links• A glossary of definitions in topology

Chapter 22

List of mathematical jargon

The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount ofjargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon oftenappears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much ofthis is common English, but with a specific non-obvious meaning when used in a mathematical sense.Some phrases, like “in general”, appear below in more than one section.

22.1 Philosophy of mathematicsabstract nonsense Also general abstract nonsense or generalized abstract nonsense, a tongue-in-cheek reference to

category theory, using which one can employ arguments that establish a (possibly concrete) result withoutreference to any specifics of the present problem.

[The paper of Eilenberg and Mac Lane (1942)] introduced the very abstract idea of a 'category' —a subject then called 'general abstract nonsense'!— Saunders Mac Lane (1997)

[Grothendieck] raised algebraic geometry to a new level of abstraction...if certain mathematicianscould console themselves for a time with the hope that all these complicated structures were 'abstract non-sense'...the later papers of Grothendieck and others showed that classical problems...which had resistedefforts of several generations of talented mathematicians, could be solved in terms of...complicated con-cepts.— Michael Monastyrsky (2001)

canonical A reference to a standard or choice-free presentation of some mathematical object. The term canonical isalso used more informally, meaning roughly “standard” or “classic”. For example, one might say that Euclid'sproof is the “canonical proof” of the infinitude of primes.

There are two canonical proofs that are always used to show non-mathematicians what a mathemat-ical proof is like:

• —The proof that there are infinitely many prime numbers.• —The proof of the irrationality of the square root of two.

— Freek Wiedijk (2006, p.2)

deep A result is called “deep” if its proof requires concepts and methods that are advanced beyond the conceptsneeded to formulate the result. The prime number theorem, proved with techniques from complex analysis,

197

198 CHAPTER 22. LIST OF MATHEMATICAL JARGON

was thought to be a deep result until elementary proofs were found. The fact that π is irrational is a deep resultbecause it requires considerable development of real analysis to prove, even though it can be stated in terms ofsimple number theory and geometry.

elegant Also beautiful; an aesthetic term referring to the ability of an idea to provide insight into mathematics,whether by unifying disparate fields, introducing a new perspective on a single field, or providing a techniqueof proof which is either particularly simple, or captures the intuition or imagination as to why the result it provesis true. Gian-Carlo Rota distinguished between elegance of presentation and beauty of concept, saying that forexample, some topics could be written about elegantly although the mathematical content is not beautiful, andsome theorems or proofs are beautiful but may be written about inelegantly.

The beauty of a mathematical theory is independent of the aesthetic qualities...of the theory’s rig-orous expositions. Some beautiful theories may never be given a presentation which matches theirbeauty....Instances can also be found of mediocre theories of questionable beauty which are given bril-liant, exciting expositions....[Category theory] is rich in beautiful and insightful definitions and poor inelegant proofs....[The theorems] remain clumsy and dull....[Expositions of projective geometry] vied forone another in elegance of presentation and in cleverness of proof....In retrospect, one wonders what allthe fuss was about.Mathematicians may say that a theorem is beautiful when they really mean to say that it is enlightening.We acknowledge a theorem’s beauty when we see how the theorem 'fits’ in its place....We say that a proofis beautiful when such a proof finally gives away the secret of the theorem....— Gian-Carlo Rota (1977, pp.173–174, pp.181–182)

elementary A proof or result is called “elementary” if it requires only basic concepts and methods, in contrast to so-called deep results. The concept of “elementary proof” is used specifically in number theory, where it usuallyrefers to a proof that does not use methods from complex analysis.

folklore A result is called “folklore” if it is non-obvious, has not been published, and yet is generally known amongthe specialists in a field. Usually, it is unknown who first obtained the result. If the result is important, it mayeventually find its way into the textbooks, whereupon it ceases to be folklore.

Many of the results mentioned in this paper should be considered “folklore” in that they merely for-mally state ideas that are well-known to researchers in the area, but may not be obvious to beginners andto the best of my knowledge do not appear elsewhere in print.— Russell Impagliazzo (1995)

natural Similar to “canonical” but more specific, this term makes reference to a description (almost exclusively inthe context of transformations) which holds independently of any choices. Though long used informally, thisterm has found a formal definition in category theory.

pathological An object behaves pathologically (or, somewhat more broadly used, in a degenerated way) if it failsto conform to the generic behavior of such objects, fails to satisfy certain regularity properties (depending oncontext), or simply disobeys mathematical intuition. These can be and often are contradictory requirements.Sometimes the term is more pointed, referring to an object which is specifically and artificially exhibited as acounterexample to these properties.

Since half a century we have seen arise a crowd of bizarre functions which seem to try to resembleas little as possible the honest functions which serve some purpose....Nay more, from the logical pointof view, it is these strange functions which are the most general....to-day they are invented expressly toput at fault the reasonings of our fathers....— Henri Poincaré (1913)

[The Dirichlet function] took on an enormous importance...as giving an incentive for the creation ofnew types of function whose properties departed completely from what intuitively seemed admissible. Acelebrated example of such a so-called 'pathological' function...is the one provided byWeierstrass....This

22.2. DESCRIPTIVE INFORMALITIES 199

function is continuous but not differentiable.— J. Sousa Pinto (2004)

Note for that latter quote that as the differentiable functions are meagre in the space of continuous functions, asBanach found out in 1931, differentiable functions are colloquially speaking a rare exception among the con-tinuous ones. Thus it can hardly be defended any-more to call non-differentiable continuous functions patho-logical.

rigor (rigour) Mathematics strives to establish its results using indisputable logic rather than informal descriptiveargument. Rigor is the use of such logic in a proof.

well-behaved An object is well-behaved (in contrast with being pathological) if it does satisfy the prevailing regularityproperties, or sometimes if it conforms to intuition (but intuition often suggests the opposite behavior as well).

22.2 Descriptive informalities

Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use de-scriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note thatmany of the terms are completely rigorous in context.

almost all A shorthand term for “all except for a set of measure zero", when there is a measure to speak of. Forexample, “almost all real numbers are transcendental" because the algebraic real numbers form a countablesubset of the real numbers with measure zero. One can also speak of “almost all” integers having a propertyto mean “all but finitely many”, despite the integers not admitting a measure for which this agrees with theprevious usage. For example, “almost all prime numbers are odd”. There is a more complicated meaning forintegers as well, discussed in the main article. Finally, this term is sometimes used synonymously with generic,below.

arbitrarily large Notions which arise mostly in the context of limits, referring to the recurrence of a phenomenonas the limit is approached. A statement such as that predicate P is satisfied by arbitrarily large values, can beexpressed in more formal notation by ∀x : ∃y ≥ x : P(y). See also frequently. The statement that quantity f(x)depending on x “can be made” arbitrarily large, corresponds to ∀y : ∃x : f(x) ≥ y.

arbitrary A shorthand for the universal quantifier. An arbitrary choice is one which is made unrestrictedly, oralternatively, a statement holds of an arbitrary element of a set if it holds of any element of that set. Also muchin general-language use among mathematicians: “Of course, this problem can be arbitrarily complicated”.

eventually, definitely In the context of limits, this is shorthand for sufficiently large arguments; the relevant argu-ment(s) are implicit in the context. As an example, one could say that “The function log(log(x)) eventuallybecomes larger than 100"; in this context, “eventually” means “for sufficiently large x".

factor through A term in category theory referring to composition of morphisms. If we have three objects A, B,and C and a map f : A → C which is written as a composition f = h ◦ g with g : A → B and h : B → C ,then f is said to factor through any (and all) of B , g , and h .

finite Next to the usual meaning of “not infinite”, in another more restrictive meaning that one may encounter, avalue being said to be “finite” also excludes infinitesimal values and the value 0. For example, if the varianceof a random variable is said to be finite, this implies it is a positive real number.

frequently In the context of limits, this is shorthand for arbitrarily large arguments and its relatives; as with eventually,the intended variant is implicit. As an example, one could say that “The function sin(x) is frequently zero”,where “frequently” means “for arbitrarily large x".

generic This term has similar connotations as almost all but is used particularly for concepts outside the purview ofmeasure theory. A property holds “generically” on a set if the set satisfies some (context-dependent) notionof density, or perhaps if its complement satisfies some (context-dependent) notion of smallness. For example,a property which holds on a dense Gδ (intersection of countably many open sets) is said to hold generically.In algebraic geometry, one says that a property of points on an algebraic variety that holds on a dense Zariskiopen set is true generically; however, it is usually not said that a property which holds merely on a dense set(which is not Zariski open) is generic in this situation.

200 CHAPTER 22. LIST OF MATHEMATICAL JARGON

in general In a descriptive context, this phrase introduces a simple characterization of a broad class of objects, withan eye towards identifying a unifying principle. This term introduces an “elegant” description which holds for"arbitrary" objects. Exceptions to this description may be mentioned explicitly, as "pathological" cases.

Norbert A’Campo of the University of Basel once asked Grothendieck about something related tothe Platonic solids. Grothendieck advised caution. The Platonic solids are so beautiful and so excep-tional, he said, that one cannot assume such exceptional beauty will hold in more general situations.— Allyn Jackson (2004, p.1197)

left-hand side, right-hand side (LHS, RHS) Most often, these refer simply to the left-hand or the right-hand sideof an equation; for example, x = y + 1 has x on the LHS and y + 1 on the RHS. Occasionally, these are usedin the sense of lvalue and rvalue: an RHS is primitive, and an LHS is derivative.

nice A mathematical object is colloquially called nice or sufficiently nice if it satisfies hypotheses or properties,sometimes unspecified or even unknown, that are especially desirable in a given context. It is an informalantonym for pathological. For example, one might conjecture that a differential operator ought to satisfy acertain boundedness condition “for nice test functions,” or one might state that some interesting topologicalinvariant should be computable “for nice spaces X.”

onto A function (which in mathematics is generally defined as mapping the elements of one set A to elements ofanother B) is called “A onto B” (instead of “A to B”) only if it is surjective; it may even be said that “f is onto”(i. e. surjective). Not translatable (without circumlocutions) to languages other than English.

proper If, for some notion of substructure, objects are substructures of themselves (that is, the relationship is re-flexive), then the qualification proper requires the objects to be different. For example, a proper subset of a setS is a subset of S that is different from S, and a proper divisor of a number n is a divisor of n that is differentfrom n. This overloaded word is also non-jargon for a proper morphism.

regular A function is called regular if it satisfies satisfactory continuity and differentiability properties, which areoften context-dependent. These properties might include possessing a specified number of derivatives, withthe function and its derivatives exhibiting some nice property, such as Hölder continuity. Informally, this termis sometimes used synonymously with smooth, below. These imprecise uses of the word regular are not to beconfused with the notion of a regular topological space, which is rigorously defined.

resp. (Respectively) A convention to shorten parallel expositions. “A (resp. B) [has some relationship to] X (resp.Y)" means that A [has some relationship to] X and also that B [has (the same) relationship to] Y. For example,squares (resp. triangles) have 4 sides (resp. 3 sides); or compact (resp. Lindelöf) spaces are ones where everyopen cover has a finite (resp. countable) open subcover.

sharp Often, a mathematical theorem will establish constraints on the behavior of some object; for example, afunction will be shown to have an upper or lower bound. The constraint is sharp (sometimes optimal) if itcannot be made more restrictive without failing in some cases. For example, for arbitrary nonnegative realnumbers x, the exponential function ex, where e = 2.7182818..., gives an upper bound on the values of thequadratic function x2. This is not sharp; the gap between the functions is everywhere at least 1. Among theexponential functions of the form αx, setting α = e2/e = 2.0870652... results in a sharp upper bound; the slightlysmaller choice α = 2 fails to produce an upper bound, since then α3 = 8 < 32. In applied fields the word “tight”is often used with the same meaning.[1]

smooth Smoothness is a concept which mathematics has endowed with many meanings, from simple differentiabilityto infinite differentiability to analyticity, and still others which are more complicated. Each such usage attemptsto invoke the physically intuitive notion of smoothness.

strong, stronger A theorem is said to be strong if it deduces restrictive results from general hypotheses. One cel-ebrated example is Donaldson’s theorem, which puts tight restraints on what would otherwise appear to be alarge class of manifolds. This (informal) usage reflects the opinion of the mathematical community: not onlyshould such a theorem be strong in the descriptive sense (below) but it should also be definitive in its area. Atheorem, result, or condition is further called stronger than another one if a proof of the second can be easilyobtained from the first. An example is the sequence of theorems: Fermat’s little theorem, Euler’s theorem,Lagrange’s theorem, each of which is stronger than the last; another is that a sharp upper bound (see above) isa stronger result than a non-sharp one. Finally, the adjective strong or the adverb strongly may be added to a

22.3. PROOF TERMINOLOGY 201

mathematical notion to indicate a related stronger notion; for example, a strong antichain is an antichain sat-isfying certain additional conditions, and likewise a strongly regular graph is a regular graph meeting strongerconditions. When used in this way, the stronger notion (such as “strong antichain”) is a technical term witha precisely defined meaning; the nature of the extra conditions cannot be derived from the definition of theweaker notion (such as “antichain”).

sufficiently large, suitably small, sufficiently close In the context of limits, these terms refer to some (unspecified,even unknown) point at which a phenomenon prevails as the limit is approached. A statement such as thatpredicate P holds for sufficiently large values, can be expressed in more formal notation by ∃x : ∀y ≥ x : P(y).See also eventually.

upstairs, downstairs A descriptive term referring to notation in which two objects are written one above the other;the upper one is upstairs and the lower, downstairs. For example, in a fiber bundle, the total space is oftensaid to be upstairs, with the base space downstairs. In a fraction, the numerator is occasionally referred to asupstairs and the denominator downstairs, as in “bringing a term upstairs”.

up to, modulo, mod out by An extension to mathematical discourse of the notions of modular arithmetic. A state-ment is true up to a condition if the establishment of that condition is the only impediment to the truth ofthe statement. Also used when working with members of equivalence classes, esp. in category theory, wherethe equivalence relation is (categorical) isomorphism; for example, “The tensor product in a weak monoidalcategory is associative and unital up to a natural isomorphism.”

vanish To assume the value 0. For example, “The function sin(x) vanishes for those values of x that are integermultiples of π.” This can also apply to limits: see Vanish at infinity.

weak, weaker The converse of strong.

well-defined Accurately and precisely described or specified.

22.3 Proof terminology

The formal language of proof draws repeatedly from a small pool of ideas, many of which are invoked through variouslexical shorthands in practice.

aliter An obsolescent term which is used to announce to the reader an alternative method, or proof of a result. In aproof, it therefore flags a piece of reasoning that is superfluous from a logical point of view, but has some otherinterest.

by way of contradiction (BWOC), or “for, if not, ...” The rhetorical prelude to a proof by contradiction, preced-ing the negation of the statement to be proved. Also, starting a proof or a sub-proof with Assume... indicatesthat a proof by contradiction will be employed.

if and only if (iff) An abbreviation for logical equivalence of statements.

in general In the context of proofs, this phrase is often seen in induction arguments when passing from the basecase to the “induction step”, and similarly, in the definition of sequences whose first few terms are exhibited asexamples of the formula giving every term of the sequence.

necessary and sufficient A minor variant on “if and only if"; “A is necessary (sufficient) for B” means “A if (onlyif) B”. For example, “For a field K to be algebraically closed it is necessary and sufficient that it have no finitefield extensions" means "K is algebraically closed if and only if it has no finite extensions”. Often used in lists,as in “The following conditions are necessary and sufficient for a field to be algebraically closed...”.

need to show (NTS), required to prove (RTP), wish to show, want to show (WTS)Proofs sometimes proceed by enumerating several conditions whose satisfaction will together imply the desiredtheorem; thus, one needs to show just these statements.

one and only one A statement of the uniqueness of an object; the object exists, and furthermore, no other suchobject exists.

202 CHAPTER 22. LIST OF MATHEMATICAL JARGON

Q.E.D. (Quod erat demonstrandum): A Latin abbreviation, meaning “which was to be demonstrated”, historicallyplaced at the end of proofs, but less common currently, having been supplanted by the Halmos end-of-proofmark.

sufficiently nice A condition on objects in the scope of the discussion, to be specified later, that will guarantee thatsome stated property holds for them. When working out a theorem, the use of this expression in the statementof the theorem indicates that the conditions involved may be not yet known to the speaker, and that the intentis to collect the conditions that will be found to be needed in order for the proof of the theorem to go through.

the following are equivalent (TFAE) Often several equivalent conditions (especially for a definition, such as normalsubgroup) are equally useful in practice; one introduces a theorem stating an equivalence of more than twostatements with TFAE.

transport of structure It is often the case that two objects are shown to be equivalent in some way, and that oneof them is endowed with additional structure. Using the equivalence, we may define such a structure on thesecond object as well, via transport of structure. For example, any two vector spaces of the same dimensionare isomorphic; if one of them is given an inner product and if we fix a particular isomorphism, then we maydefine an inner product on the other space by factoring through the isomorphism.

Let V be a finite-dimensional vector space over k....Let (ei)₁ ≤ i ≤ n be a basis for V....There is anisomorphism of the polynomial algebra k[Tij]₁ ≤ i,j ≤ n onto the algebra Symk(V ⊗ V*)....It extendsto an isomorphism of k[GLn] to the localized algebra Symk(V ⊗ V*)D, where D = det(ei ⊗ ej*)....Wewrite k[GL(V)] for this last algebra. By transport of structure, we obtain a linear algebraic groupGL(V)isomorphic to GLn.— Igor Shafarevich (1991, p.12)

without (any) loss of generality (WLOG, WOLOG, WALOG), we may assume (WMA)Sometimes a proposition can be more easily proved with additional assumptions on the objects it concerns. Ifthe proposition as stated follows from this modified one with a simple and minimal explanation (for example, ifthe remaining special cases are identical but for notation), then the modified assumptions are introduced withthis phrase and the altered proposition is proved.

22.4 Proof techniques

Mathematicians have several phrases to describe proofs or proof techniques. These are often used as hints for fillingin tedious details.

angle chasing Used to describe a geometrical proof that involves finding relationships between the various anglesin a diagram.[2]

back-of-the-envelope calculation An informal computation omitting much rigor without sacrificing correctness.Often this computation is “proof of concept” and treats only an accessible special case.

by inspection A rhetorical shortcut made by authors who invite the reader to verify, at a glance, the correctnessof a proposed expression or deduction. If an expression can be evaluated by straightforward application ofsimple techniques and without recourse to extended calculation or general theory, then it can be evaluated byinspection. It is also applied to solving equations; for example to find roots of a quadratic equation by inspectionis to 'notice' them, or mentally check them. 'By inspection' can play a kind of gestalt role: the answer or solutionsimply clicks into place.

clearly, can be easily shown A term which shortcuts around calculation the mathematician perceives to be tediousor routine, accessible to any member of the audience with the necessary expertise in the field; Laplace usedobvious (French: évident).

complete intuition commonly reserved for jokes (puns on complete induction).

22.5. NOTES 203

diagram chasing [3] Given a commutative diagram of objects and morphisms between them, if one wishes to provesome property of the morphisms (such as injectivity) which can be stated in terms of elements, then the proofcan proceed by tracing the path of elements of various objects around the diagram as successive morphismsare applied to it. That is, one chases elements around the diagram, or does a diagram chase.

handwaving A non-technique of proof mostly employed in lectures, where formal argument is not strictly necessary.It proceeds by omission of details or even significant ingredients, and is merely a plausibility argument.

in general In a context not requiring rigor, this phrase often appears as a labor-saving device when the technicaldetails of a complete argument would outweigh the conceptual benefits. The author gives a proof in a simpleenough case that the computations are reasonable, and then indicates that “in general” the proof is similar.

index battle for proofs involving object with plurious indices which can be solved by going to the bottom (if anyonewishes to take up the effort). Similar to diagram chasing.

trivial Similar to clearly. A concept is trivial if it holds by definition, is immediate corollary to a known statement,or is a simple special case of a more general concept.

22.5 Notes[1] Boyd, Stephen (2004). Convex Optimization. Cambridge University Press. ISBN 0521833787.

[2] Roe, John (1993), Elementary Geometry, Oxford science publications, p. 119, ISBN 0-19-853456-6

[3] Numerous examples can be found in (Mac Lane 1998), for example on p. 100.

22.6 References• Eilenberg, Samuel; Mac Lane, Saunders (1942), “Natural Isomorphisms in Group Theory”, Proc. Natl. Acad.Sci. USA 28: 537–543, doi:10.1073/pnas.28.12.537.

• Impagliazzo, Russell (1995), “A personal view of average-case complexity”, Proc. Tenth Annual Structure inComplexity Theory Conference (SCT'95), pp. 134–147, doi:10.1109/SCT.1995.514853.

• Jackson, Allyn (2004), “Comme Appelé du Néant — As If Summoned from the Void: The Life of AlexandreGrothendieck”, AMS Notices 51 (9,10) (Parts I and II).

• Mac Lane, Saunders (1997), “The PNAS way back then” (PDF), Proc. Natl. Acad. Sci. USA 94 (12): 5983–5985, doi:10.1073/pnas.94.12.5983.

• Mac Lane, Saunders (1998), Categories for the Working Mathematician, Springer.

• Monastyrsky, Michael (2001), “Some Trends in Modern Mathematics and the Fields Medal” (PDF), Can.Math. Soc. Notes 33 (2 and 3).

• Pinto, J. Sousa (2004), Hoskins, R.F., ed., Infinitesimal methods for mathematical analysis, Horwood Publish-ing, p. 246, ISBN 978-1-898563-99-0.

• Poincare, Henri (1913), Halsted, Bruce, ed., The Foundations of Science, The Science Press, p. 435.

• Rota, Gian-Carlo (1977), “The phenomenology ofmathematical beauty”, Synthese 111 (2): 171–182, doi:10.1023/A:1004930722234,ISSN 0039-7857.

• Shafarevich, Igor (1991), Kandall, G.A., ed., Algebraic Geometry IV, Springer.

• Wiedijk, Freek, ed. (2006), The Seventeen Provers of the World, Birkhäuser, ISBN 3-540-30704-4.

204 CHAPTER 22. LIST OF MATHEMATICAL JARGON

22.7 Text and image sources, contributors, and licenses

22.7.1 Text• Glossary of areas of mathematics Source: https://en.wikipedia.org/wiki/Glossary_of_areas_of_mathematics?oldid=685378455 Con-tributors: Zundark, Michael Hardy, Topbanana, Jayjg, Chenxlee, Sodin, Wavelength, Joel7687, Xaxafrad, John, Myasuda, Cydebot,JaGa, Davidmanheim, R'n'B, Flyer22 Reborn, Niceguyedc, Arjayay, SchreiberBike, Offbeatcinema, Ozob, Yobot, LilHelpa, FrescoBot,MegaSloth, John of Reading, D.Lazard, SporkBot, DPL bot, Killikalli, Brad7777, The1337gamer, LegacyOfValor, Mandm900 andAnonymous: 4

• Glossary of arithmetic and Diophantine geometry Source: https://en.wikipedia.org/wiki/Glossary_of_arithmetic_and_Diophantine_geometry?oldid=684259508 Contributors: Michael Hardy, Charles Matthews, Gandalf61, Giftlite, Bender235, Ruud Koot, Timen-dum, Rjwilmsi, Mathbot, JosephSilverman, Tesseran, Valoem, George100, CRGreathouse, Cydebot, Vanish2, Diotime, CopyToWik-tionaryBot, Tearsinraine, Remember the dot, VolkovBot, CohesionBot, Addbot, Lightbot, Yobot, FactSpewer, Charvest, RjwilmsiBot,MithrandirAgain, Suslindisambiguator, SporkBot, Helpful Pixie Bot, Brad7777, Deltahedron, Spectral sequence, Monkbot and Anony-mous: 10

• Glossary of classical algebraic geometry Source: https://en.wikipedia.org/wiki/Glossary_of_classical_algebraic_geometry?oldid=680222247Contributors: Michael Hardy, TakuyaMurata, Giftlite, Rgdboer, Rjwilmsi, R.e.b., Wavelength, Gaius Cornelius, SMcCandlish, Load-master, David Eppstein, Robertgreer, SchreiberBike, Yobot, AnomieBOT, LilHelpa, FrescoBot, Trappist the monk, John of Reading,D.Lazard, BTotaro, Jochen Burghardt and Anonymous: 2

• Glossary of cryptographic keys Source: https://en.wikipedia.org/wiki/Glossary_of_cryptographic_keys?oldid=559608936 Contribu-tors: Michael Hardy, Securiger, UtherSRG, Alan Liefting, Alensha, Matt Crypto, Sam Hocevar, TheObtuseAngleOfDoom, ArnoldRein-hold, Reinyday, R. S. Shaw, Thryduulf, Woohookitty, Kbdank71, Dpv, Josh Parris, The wub, Intgr, SMcCandlish, Rwwww, SmackBot,GBL, Unint, PrimeHunter, Xyzzyplugh, Ohconfucius, CenozoicEra, Cydebot, GromXXVII, Greensburger, CopyToWiktionaryBot, Clue-Bot NG and Anonymous: 6

• Glossary of differential geometry and topology Source: https://en.wikipedia.org/wiki/Glossary_of_differential_geometry_and_topology?oldid=659073454 Contributors: Michael Hardy, Dcljr, Charles Matthews, Dysprosia, Tosha, Mikez, Fropuff, Waltpohl, CryptoDerk, D6,Rgdboer, Oleg Alexandrov, Ruud Koot, Mathbot, Archelon, SmackBot, Foxjwill, Xyzzyplugh, Cydebot, David Eppstein, CopyToWik-tionaryBot, Katzmik, Jasonyo, Versus22, Yobot, Point-set topologist, Erik9bot, Rausch, SporkBot, Brad7777 and Anonymous: 9

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• Glossary of field theory Source: https://en.wikipedia.org/wiki/Glossary_of_field_theory?oldid=682241092 Contributors: AxelBoldt,Michael Hardy, Wshun, Dcljr, Loren Rosen, CharlesMatthews, Dysprosia, Robbot, Fropuff, D6, Rich Farmbrough, Ruud Koot, Marudub-shinki, BD2412, Marco Streng, Dmharvey, SmackBot, Xyzzyplugh, Cydebot, Nick Number, STBot, CopyToWiktionaryBot, Barylior,Addbot, Bte99, Yobot, Erik9bot, Fred Gandt, CaroleHenson, Solomon7968, Deltahedron and Anonymous: 8

• Glossary of game theory Source: https://en.wikipedia.org/wiki/Glossary_of_game_theory?oldid=558243640 Contributors: MichaelHardy, Samw, Mousomer, Avihu, Kappa, Tabor, Kzollman, Ruud Koot, Rlove, SmackBot, Xyzzyplugh, Amakuru, CmdrObot, Cyde-bot, Jeff.kowalski, Jakob.scholbach, Hlprasad~enwiki, David Eppstein, Gwern, CopyToWiktionaryBot, Philip Trueman, Dan Polansky,ClueBot, Addbot, Ilikebeansandweiners, Erik9bot and Anonymous: 14

• Glossary of graph theory Source: https://en.wikipedia.org/wiki/Glossary_of_graph_theory?oldid=685400754 Contributors: DamianYerrick, XJaM, Nonenmac, Tomo, Edward, Patrick, Michael Hardy, Wshun, Booyabazooka, Dcljr, TakuyaMurata, GTBacchus, Eric119,CharlesMatthews, Dcoetzee, Dysprosia, Doradus, Reina riemann, Markhurd, Maximus Rex, Hyacinth, Populus, Altenmann, MathMartin,Bkell, Giftlite, Dbenbenn, Brona, Sundar, GGordonWorleyIII, HorsePunchKid, Peter Kwok, D6, Rich Farmbrough, ArnoldReinhold, PaulAugust, Bender235, Zaslav, Kjoonlee, Elwikipedista~enwiki, El C, Yitzhak, TheSolomon, A1kmm, 3mta3, Jérôme, Ricky81682, Rd-vdijk, Oleg Alexandrov, Joriki, Linas, MattGiuca, Ruud Koot, Jwanders, Xiong, Lasunncty, SixWingedSeraph, Grammarbot, Tizio, Salixalba, Mathbot, Margosbot~enwiki, Sunayana, Pojo, Quuxplusone, Vonkje, N8wilson, Chobot, Algebraist, YurikBot, Me and, Altoid,Grubber, Archelon, Gaius Cornelius, Rick Norwood, Ott2, Closedmouth, SmackBot, Stux, Achab, Brick Thrower, Mgreenbe, Mcld,Michael@nosivad.com, Lansey, Thechao, JLeander, DVanDyck, Quaeler, RekishiEJ, CmdrObot, Csaracho, Citrus538, Jokes Free4Me,Cydebot, Starcrab, Quintopia, Ferris37, Scarpy, Headbomb, Salgueiro~enwiki, Spanningtree, A3nm, David Eppstein, JoergenB, Kope,AlexShkotin, CopyToWiktionaryBot, R'n'B, Leyo, Mikhail Dvorkin, The Transliterator, Ratfox, MentorMentorum, Skaraoke, Sani-tySolipsism, Anonymous Dissident, PaulTanenbaum, Ivan Štambuk, Whorush, Eggwadi, Thehotelambush, Doc honcho, Anchor LinkBot, Rsdetsch, Denisarona, Justin W Smith, Unbuttered Parsnip, Happynomad, Alexey Muranov, Addbot, Aarond144, Jfitzell, NateWessel, Yobot, Jalal0, Ian Kelling, Citation bot, Buenasdiaz, Twri, Kinewma, Miym, Prunesqualer, Mzamora2, JZacharyG, Pmq20,Shadowjams, Hobsonlane, DixonDBot, Reaper Eternal, EmausBot, John of Reading, Wikipelli, Bethnim, Mastergreg82, ClueBot NG,EmanueleMinotto, Warumwarum, DavidRideout, BG19bot, Andrey.gric, Szebenisz, BattyBot, Eduardofeld, ChrisGualtieri, Deltahedron,Jw489kent, Jmerm, Morgoth106, SofjaKovalevskaja and Anonymous: 137

• Glossary of invariant theory Source: https://en.wikipedia.org/wiki/Glossary_of_invariant_theory?oldid=680222569Contributors: Rjwilmsi,Salix alba, R.e.b., Wavelength, SMcCandlish, Gilo1969, Trappist the monk and John of Reading

• Glossary of Lie algebras Source: https://en.wikipedia.org/wiki/Glossary_of_Lie_algebras?oldid=687735620 Contributors: WilliamAvery, Michael Hardy, TakuyaMurata, Charles Matthews, Topbanana, Rich Farmbrough, Burn, RHaworth, Ketiltrout, R.e.b., MichaelKinyon, CmdrObot, Cydebot, R'n'B, Daniel5Ko, Hans Adler, SchreiberBike, Yobot, Worldbruce, FrescoBot, Luizpuodzius, Brad7777,Ilyats, SJ Defender and Anonymous: 7

• Glossary ofmodule theory Source: https://en.wikipedia.org/wiki/Glossary_of_module_theory?oldid=674675066Contributors: Rgdboer,Burn, Bejnar, Rschwieb, Cydebot, YohanN7, JackSchmidt, Hans Adler, Trappist the monk, Brad7777, BattyBot, Escspeed and Anony-mous: 2

• Glossary of order theory Source: https://en.wikipedia.org/wiki/Glossary_of_order_theory?oldid=625139014 Contributors: AxelBoldt,Zundark, Toby Bartels, Patrick, Michael Hardy, Dcljr, Charles Matthews, Malcohol, Tobias Bergemann, Weialawaga~enwiki, Markus

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Krötzsch, Waltpohl, D6, Noisy, Paul August, Zaslav, S.K., EmilJ, Oleg Alexandrov, Ruud Koot, Jeff3000, MZMcBride, Hairy Dude,Trovatore, Xyzzyplugh, Ohconfucius, Cesium 133, Cydebot, Mojo Hand, RobHar, Vanish2, MetsBot, CopyToWiktionaryBot, R'n'B,Reedy Bot, Cmbankester, Smarccohen, PaulTanenbaum, StevenJohnston, Marc van Leeuwen, Yobot, Pcap, R.tahiry, Mwenyendoto,Twri, SporkBot, Helpful Pixie Bot, Deltahedron, Metadox and Anonymous: 7

• Glossary ofRiemannian andmetric geometry Source: https://en.wikipedia.org/wiki/Glossary_of_Riemannian_and_metric_geometry?oldid=650225552 Contributors: AxelBoldt, Michael Hardy, Dcljr, TakuyaMurata, Darkwind, Nikai, Charles Matthews, Jitse Niesen,Bkell, Acm, Tosha, Mikez, Lethe, Fropuff, Beland, Chmod007, D6, Number 0, SteinbDJ, Oleg Alexandrov, Linas, Ruud Koot, Salixalba, Mathbot, Gurch, KSmrq, Trovatore, SmackBot, JFManning, Nbarth, Jhausauer, Xyzzyplugh, Tesseran, Eassin, Cydebot, NickNum-ber, Quarague, Sullivan.t.j, David Eppstein, Squidonius, CopyToWiktionaryBot, Lantonov, VolkovBot, Commentor, Arcfrk, Katzmik,KoenDelaere, Anchor Link Bot, Haroldsultan, Jjauregui, Addbot, Gejian, Erik Streb, Yobot, Howard McCay, Erik9bot, Captain-n00dle,Jfdavis, SporkBot, Brad7777 and Anonymous: 19

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• Glossary of semisimple groups Source: https://en.wikipedia.org/wiki/Glossary_of_semisimple_groups?oldid=662879967 Contribu-tors: Charles Matthews, Ruud Koot, SmackBot, Colonies Chris, Lambiam, RichardF, Cydebot, RobHar, Magioladitis, CopyToWik-tionaryBot, Yobot, Erik9bot, DivineAlpha, Zenkunen, SporkBot, ClueBot NG, Mark L MacDonald and

• Glossary of shapes with metaphorical names Source: https://en.wikipedia.org/wiki/Glossary_of_shapes_with_metaphorical_names?oldid=680558588 Contributors: Dominus, Altenmann, Jason Quinn, Beland, Pmanderson, Urhixidur, Brianhe, DiegoMoya, Oleg Alexan-drov, Salix alba, R.e.b., Srleffler, Gaius Cornelius, Anomalocaris, Grafen, Retired username, Webster100, Bibliomaniac15, Crystallina,SmackBot, RDBury, Unint, PrimeHunter, Hoof Hearted, Jaganath, JayHenry, CmdrObot, CBM, ShelfSkewed, Lurlock, Yaris678, Cyde-bot, Alfirin, Mattisse, GromXXVII,Magioladitis, Seam.us, BilCat, JaGa, CopyToWiktionaryBot, Dodger67, Martarius, ClueBot, Abrech,SchreiberBike, Fyrael, Yobot, AnomieBOT, Gestme nor u, ClueBot NG, BG19bot, HueSatLum, Frosty and Anonymous: 13

• Glossary of string theory Source: https://en.wikipedia.org/wiki/Glossary_of_string_theory?oldid=680223137 Contributors: R.e.b.,Wavelength, SMcCandlish, Chris the speller, Colonies Chris, Bearian, Sun Creator, Tom.Reding, TedderBot, John of Reading, Maschen,Soni and Mmitchell10

• Glossary of systems theory Source: https://en.wikipedia.org/wiki/Glossary_of_systems_theory?oldid=614232115 Contributors: Mdd,Ceyockey, Woohookitty, Ruud Koot, BD2412, RussBot, AndrewHowse, Cydebot, Magioladitis, Meredyth, Erkan Yilmaz, LittleHow,Kenneth M Burke, IPSOS, Gekritzl, DerBorg, Tassedethe, LeoLady53, J04n, SporkBot, PhnomPencil and Anonymous: 9

• Glossary of tensor theory Source: https://en.wikipedia.org/wiki/Glossary_of_tensor_theory?oldid=666163224 Contributors: MichaelHardy, Dcljr, Charles Matthews, Hankwang, Marc Venot, Lockeownzj00, Beland, D6, Pjacobi, ArnoldReinhold, Neonumbers, OlegAlexandrov, RuudKoot, Grammarbot, DanMS, SmackBot, TimBentley, Nbarth, Colonies Chris, Xyzzyplugh, Cydebot, OinkOink~enwiki,David Eppstein, CopyToWiktionaryBot, Lantonov, LokiClock, Geometry guy, Barak Sh, Yobot, J04n, Omnipaedista, Erik9bot, Ameten,Quondum, Maschen, Mgvongoeden, F=q(E+v^B), Monkbot and Anonymous: 7

• Glossary of topology Source: https://en.wikipedia.org/wiki/Glossary_of_topology?oldid=686937092 Contributors: AxelBoldt, Mav,Zundark, Toby~enwiki, Toby Bartels, PierreAbbat, Michael Hardy, Wshun, Dineshjk, Dcljr, TakuyaMurata, Hashar, Loren Rosen, Re-volver, Charles Matthews, Dcoetzee, Dysprosia, MathMartin, Anthony, Tobias Bergemann, Tosha, Giftlite, Markus Krötzsch, Nunh-huh,Lethe, Fropuff, Beland, Sam Hocevar, Paul August, EmilJ, C S, Tsirel, Kuratowski’s Ghost, Msh210, Olegalexandrov, Oleg Alexandrov,Firsfron, Linas, Pol098, Ruud Koot, Ryan Reich, Graham87, BD2412, Dpv, Margosbot~enwiki, CiaPan, YurikBot, Eraserhead1, Trova-tore, Number 57, Kompik, Sardanaphalus, SmackBot, Incnis Mrsi, Acipsen, Xyzzyplugh, Dreadstar, Lambiam, Jim.belk, Texas Dervish,Myasuda, Gregbard, Cydebot, Krauss, WinBot, Faizhaider, Wlod, Bwhack, David Eppstein, CopyToWiktionaryBot, R'n'B, Adavidb,2help, Joshua Issac, Fuzzygenius, VolkovBot, Anchor Link Bot, Cenarium, Hans Adler, Jaan Vajakas, Addbot, Download, A:-)Brunuś,Yobot, LilHelpa, DSisyphBot, Phn229, Erik9bot, FrescoBot, Åkebråke, Junior Wrangler, WikitanvirBot, Irina Gelbukh, SporkBot, Help-ful Pixie Bot, Paolo Lipparini, AvocatoBot, Brad7777, BattyBot, Deltahedron, Spectral sequence, Blue cheese mite, Natural boundaryand Anonymous: 32

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Glossary of set theoryFrom Wikipedia, the free encyclopedia

This is a glossary of set theory.

Contents :Greek · !$@ · A · B · C · D · E · F · G · H · I · J · K · L · M · N · O · P · Q · R · S · T · U · V · W · XYZ · See also · References

Greek

αOften used for an ordinal

ββX is the Stone–Čech compactification of X

γ

A gamma number, an ordinal of the form ωα

ΓThe Gamma function of ordinals. In particular Γ0 is the Feferman–Schütte ordinal.

δ

A delta number, an ordinal of the form ωωα

Δ (Greek capital delta, not to be confused with a triangle ∆)1. A set of formulas in the Lévy hierarchy2. A delta system

ε

An epsilon number, an ordinal with ωε=ε

η1. The order type of the rational numbers2. An eta set, a type of ordered set

θThe order type of the real numbers

κOften used for a cardinal, especially the critical point of an elementary embedding

λ1. Often used for a cardinal2. The order type of the real numbers

μA measure

Π1. A product of cardinals2. A set of formulas in the Lévy hierarchy

ρThe rank of a set

σcountable, as in σ-compact, σ-complete and so on

Σ

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1. A sum of cardinals2. A set of formulas in the Lévy hierarchy

φA Veblen function

ω1. The smallest infinite ordinal2. ωα is an alternative name for ℵα, used when it is considered as an ordinal number rather than a cardinal number

Ω1. The class of all ordinals, related to Cantor's absolute2. Ω-logic is a form of logic introduced by Hugh Woodin

!$@

∈, =, ⊆, ⊇, ⊃, ⊂, ∪, ∩, ∅Standard set theory symbols with their usual meanings (is a member of, equals, is a subset of, is a superset of, is a proper superset of, isa proper subset of, union, intersection, empty set)

∧ ∨ → ↔ ¬ ∀ ∃Standard logical symbols with their usual meanings (and, or, implies, is equivalent to, not, for all, there exists)

⨡f⨡X is the restriction of a function f to some set X

∆ (A triangle, not to be confused with the Greek letter Δ)1. The symmetric difference of two sets2. A diagonal intersection

◊The diamond principle

♣A clubsuit principle

□The square principle

∘The composition of functions

⁀s⁀x is the extension of a sequence s by x

+1. Addition of ordinals2. Addition of cardinals

3. α+ is the smallest cardinal greater than α

4. B+ is the poset of nonzero elements of a Boolean algebra B5. The inclusive or operation in a Boolean algebra. (In ring theory it is used for the exclusive or operation)

−The difference of two sets: x−y is the set of elements of x not in y.

×A product of sets

/A quotient of a set by an equivalence relation

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⋅1. x⋅y is the ordinal product of two ordinals2. x⋅y is the cardinal product of two cardinals

*An operation that takes a forcing poset and a name for a forcing poset and produces a new forcing poset.

∞The class of all ordinals, or at least something larger than all ordinals

αΒ

1. Cardinal exponentiation2. Ordinal exponentiation

Βα

1. The set of functions from Β to α

→1. Implies2. f:X→Y means f is a function from X to Y.

3. The ordinary partition symbol, where κ→(λ)nm means that for every coloring of the n-element subsets of κ with m colors there is a

subset of size λ all of whose n-element subsets are the same color.

f ' xIf there is a unique y such that ⟨x,y⟩ is in f then f ' x is y, otherwise it is the empty set. So if f is a function and x is in its domain, then f ' xis f(x).

f “ Xf “ X is the image of a set X by f. If f is a function whose domain contains X this is {f(x):x∈X}

[ ]1. M[G] is the smallest model of ZF containing G and all elements of M.

2. [α]β is the set of all subsets of a set α of cardinality Β, or of an ordered set α of order type Β3. [x] is the equivalence class of x

{ }1. {a, b, ...} is the set with elements a, b, ...2. {x : φ(x)} is the set of x such that φ(x)

⟨ ⟩⟨a,b⟩ is an ordered pair, and similarly for ordered n-tuples

|X|The cardinality of a set X

||φ||The value of a formula φ in some Boolean algebra

⌜φ⌝⌜φ⌝ (Quine quotes, unicode U+231C, U+231D) is the Gödel number of a formula φ

⊦A⊦φ means that the formula φ follows from the theory A

⊧A⊧φ means that the formula φ holds in the model A

⊩

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The forcing relation

≺An elementary embedding

⊥p⊥q means that p and q are incompatible elements of a partial order

0#

zero sharp, the set of true formulas about indiscernibles and order-indiscernibles in the constructible universe

0†

zero dagger, a certain set of true formulas

ℵThe Hebrew letter aleph, which indexes the aleph numbers or infinite cardinals ℵα

בThe Hebrew letter beth, which indexes the beth numbers αב

A serif form of the Hebrew letter gimel, representing the gimel function

תThe Hebrew letter Taw, used by Cantor for the class of all cardinal numbers

A

�

The least size of a maximal almost disjoint family of infinite subsets of ω

AThe Suslin operation

absolute1. A statement is called absolute if its truth in some model implies its truth in certain related models2. Cantor's absolute is a somewhat unclear concept sometimes used to mean the class of all sets3. Cantor's absolute infinite Ω is a somewhat unclear concept related to the class of all ordinals

AC1. AC is the Axiom of choice2. ACω is the Axiom of countable choice

ADThe axiom of determinacy

admissibleAn admissible set is a model of Kripke–Platek set theory, and an admissible ordinal is an ordinal α such that Lα is an admissible set

AHAleph hypothesis, a form of the generalized continuum hypothesis

aleph1. An infinite cardinalThe aleph function taking ordinals to infinite cardinals

almost universalA class is called almost universal if every subset of it is contained in some member of it

amenable

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An amenable set is a set that is a model of Kripke–Platek set theory without the axiom of collection

analyticAn analytic set is the continuous image of a Polish space

antichainAn antichain is a set of pairwise incompatible elements of a poset

Aronszajn treeAn Aronszajn tree is an uncountable tree such that all branches and levels are countable. More generally a κ-Aronszajn tree is a tree ofcardinality κ such that all branches and levels have cardinality less than κ

atom1. An urelement, something that is not a set but allowed to be an element of a set2. An element of a poset such that any two elements smaller than it are compatible.

axiomAczel's anti-foundation axiom states that every accessible pointed directed graph corresponds to a unique setAD+ An extension of the axiom of determinacyAxiom of adjunction Adjoining a set to another set produces a setAxiom of amalgamation The union of all elements of a set is a set. Same as axiom of unionAxiom of choice The product of any set of non-empty sets is non-emptyAxiom of collection This can mean either the axiom of replacement or the axiom of separationAxiom of comprehension The class of all sets with a given property is a set. Usually contradictory.Axiom of constructibility Any set is constructible, often abbreviated as V=LAxiom of countability Every set is hereditarily countableAxiom of countable choice The product of a countable number of non-empty sets is non-emptyAxiom of dependent choice A weak form of the axiom of choiceAxiom of determinacy Certain games are determined, in other words one player has a winning strategyAxiom of empty set The empty set existsAxiom of extensionality or axiom of extentAxiom of foundation Same as axiom of regularityAxiom of global choice There is a global choice functionAxiom of heredity (any member of a set is a set; used in Ackermann's system.)Axiom of infinity There is an infinite setAxiom of limitation of size A class is a set if and only if it has smaller cardinality than the class of all setsAxiom of pairing Unordered pairs of sets are setsAxiom of power set The powerset of any set is a setAxiom of projective determinacy Certain games given by projective set are determined, in other words one player has a winningstrategyAxiom of real determinacy Certain games are determined, in other words one player has a winning strategyAxiom of regularity Sets are well foundedAxiom of replacement The image of a set under a function is a set. Same as axiom of substitutionAxiom of subsets The powerset of a set is a set. Same as axiom of powersetsAxiom of substitution The image of a set under a function is a setAxiom of union The union of all elements of a set is a setAxiom schema of predicative separation Axiom of separation for formulas whose quantifiers are boundedAxiom schema of replacement The image of a set under a function is a setAxiom schema of separation The elements of a set with some property form a setAxiom schema of specification The elements of a set with some property form a set. Same as axiom schema of separationFreiling's axiom of symmetry is equivalent to the negation of the continuum hypothesisMartin's axiom statess very roughly that cardinals less than the cardinality of the continuum behave like ℵ0.

The proper forcing axiom is a strengthening of Martin's axiom

B

�

The bounding number, the least size of an unbounded family of sequences of natural numbers

Baire

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1. René-Louis Baire2. A subset of a topological space has the Baire property if it differs from an open set by a meager set3. The Baire space is a topological space whose points are sequences of natural numbers4. A Baire space is a topological space such that every intersection of a countable collection of open dense sets is dense

basic set theory1. Naive set theory2. A weak set theory, given by Kripke–Platek set theory without the axiom of collection

Berkeley cardinalA Berkeley cardinal is a cardinal κ in a model of ZF such that for every transitive set M that includes κ, there is a nontrivial elementaryembedding of M into M with critical point below κ.

BA Boolean algebra

BABaumgartner's axiom, one of three axioms introduced by Baumgartner.

BGBernays–Gödel set theory without the axiom of choice

BGCBernays–Gödel set theory with the axiom of choice

Borel setA Borel set is a set in the smallest sigma algebra containing the open sets

bounding numberThe bounding number is the least size of an unbounded family of sequences of natural numbers

BSBasic set theory

Burali-Forti paradoxThe Burali-Forti paradox states that the ordinal numbers do not form a set

C

c

�

The cardinality of the continuum

CThe Cantor set

Cantor1. Georg Cantor2. The Cantor normal form of an ordinal is its base ω expansion.3. Cantor's paradox says that the powerset of a set is larger than the set, which gives a contradiction when applied to the universal set.4. The Cantor set, a perfect nowhere dense subset of the real line5. Cantor's absolute infinite Ω is something to do with the class of all ordinals6. Cantor's absolute is a somewhat unclear concept sometimes used to mean the class of all sets7. Cantor's theorem states that the powerset operation increases cardinalities

cardinal1. A cardinal number is an ordinal with more elements than any smaller ordinal

cardinalityThe number of elements of a set

categorical

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A theory is called κ-categorical if all models of cardinality κ are isomorphic

category1. A set of first category is the same as a meager set: a set that is the union of a countable number of nowhere-dense sets, and a set ofsecond category is a set that is not of first category.2. A category in the sense of category theory.

ccccountable chain condition

cfThe cofinality of an ordinal

CHThe continuum hypothesis

clAbbreviation for "closure of" (a set under some collection of operations)

classA class is a collection of sets

clubA contraction of "closed unbounded"1. A club set is a closed unbounded subset, often of an ordinal2. The club filter is the filter of all subsets containing a club set3. Clubsuit is a combinatorial principle similar to but weaker than the diamond principle

cofinalA subset of a poset is called cofinal if every element of the poset is at most some element of the subset.

cofinalityThe cofinality of a poset (especially an ordinal or cardinal) is the smallest cardinality of a cofinal subset

Col

collapsing algebraA collapsing algebra Col(κ,λ) collapses cardinals between λ and κ

complete1. "Complete set" is an old term for "transitive set"2. A theory is called complete if it assigns a truth value (true or false) to every statement of its language3. An ideal is called κ-complete if it is closed under the union of less than κ elements4. A measure is called κ-complete if the union of less than κ measure 0 sets has measure 05. A linear order is called complete if every nonempty bounded subset has a least upper bound

ConCon(T) for a theory T means T is consistent

condensation lemmaGödel's condensation lemma says that an elementary submodel of an element of the construcible hierarchy is also an element of theconstructible hierarchy

constructibleA set is called constructible if it is in the constructible universe.

continuumThe continuum is the real line or its cardinality

coreA core model is a special sort of inner model generalizing the constructible universe

countable chain condition

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The countable chain condition (ccc) for a poset states that every antichain is countable

critical pointThe critical point κ of an elementary embedding j of the universe into itself is the smallest cardinal κ not mapped to itself

CTMCountable transitive model

cumulative hierarchyA cumulative hierarchy is a sequence of sets indexed by ordinals that satisfies certain conditions and whose union is used as a model ofset theory

D

�

The dominating number of a poset

DCThe axiom of dependent choice

defThe set of definable subsets of a set

definableA subset of a set is called definable set if it is the collection of elements satisfying a sentence in some given language

delta

1. A delta number is an ordinal of the form ωωα

2. A delta system, also called a sunflower, is a collection of sets such that any two distinct sets have intersection X for some fixed set X

DfThe set of definable subsets of a set

diagonal intersection

If is a sequence of subsets of an ordinal , then the diagonal intersection is

diamond principleJensen's diamond principle states that there are sets Aα⊆α for α<ω1 such that for any subset A of ω1 the set of α with A∩α = Aα is

stationary in ω1.

domThe domain of a function

E

Easton's theoremEaston's theorem describes the possible behavior of the powerset function on regular cardinals

EATSThe statement "every Aronszajn tree is special"

elementaryAn elementary embedding is a function preserving all properties describable in the language of set theory

epsilon number

An epsilon number is an ordinal α such that α=ωα

Erdos

Erdős1. Paul Erdős2. An Erdős cardinal is a large cardinal satisfying a certain partition condition

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3. The Erdős–Rado theorem extends Ramsey's theorem to infinite cardinals

ethereal cardinalAn ethereal cardinal is a type of large cardinal similar in strength to subtle cardinals

extendible cardinalA cardinal κ is called extendible if for all η there is a nontrivial elementary embedding of Vκ+η into some Vλ with critical point κ

F

FAn Fσ is a union of a countable number of closed sets

filterA filter is a non-empty subset of a poset that is downward-directed and upwards-closed

finite intersection property

FIPThe finite intersection property, abbreviated FIP, says that the intersection of any finite number of elements of a set is non-empty

first1. A set of first category is the same as a meager set: one that is the union of a countable number of nowhere-dense sets.2. An ordinal of the first class is a finite ordinal3. First order logic allows quantification over elements of a model, but not over subsets

Fodor's lemmaFodor's lemma states that a regressive function on a regular uncountable cardinal is constant on a stationary subset.

forcingForcing (set theory) is a method of adjoining a generic filter G of a poset P to a model of set theory M to obtain a new model M[G]

G

G1. A generic ultrafilter2. A Gδ is a countable intersection of open sets

gamma number

A gamma number is an ordinal of the form ωα

GCHGeneralized continuum hypothesis

generalized continuum hypothesis

The generalized continuum hypothesis states that 2ℵα = ℵα+1

generic1. A generic filter of a poset P is a filter that intersects all dense subsets of P that are contained in some model M.2. A generic extension of a model M is a model M[G] for some generic filter G.

global choiceThe axiom of global choice says there is a well ordering of the class of all sets

Godel

Gödel1. Kurt Gödel2. A Gödel number is a number assigned to a formula3. The Gödel universe is another name for the constructible universe4. Gödel's incompleteness theorems show that sufficiently powerful consistent recursively enumerable theories cannot be complete5. Gödel's completeness theorem states that consistent first-order theories have models

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H

HAbbreviation for "hereditarily"

Hκ

H(κ)The set of sets that are hereditarily of cardinality less than κ

Hartogs numberThe Hartogs number of a set X is the least ordinal α such that there is no injection from α into X.

Hausdorff gapA Hausdorff gap is a gap in the ordered set of growth rates of sequences of integers, or in a similar ordered set

HCThe set of hereditarily countable sets

hereditarilyIf P is a property the a set is hereditarily P if all elements of its transitive closure have property P. Examples: Hereditarily countable setHereditarily finite set

HFThe set of hereditarily finite sets

HSThe class of hereditarily symmetric sets

HODThe class of hereditarily ordinal definable sets

huge cardinalA huge cardinal is a cardinal number κ such that there exists an elementary embedding j : V → M with critical point κ from V into atransitive inner model M containing all sequences of length j(κ) whose elements are in M

I

I0, I1, I2, I3The rank-into-rank large cardinal axioms

idealAn ideal in the sense of ring theory, usually of a Boolean algebra, especially the Boolean algebra of subsets of a set

inaccessible cardinalAn inaccessible cardinal is a regular uncountable cardinal that is a (weak or strong) limit

indecomposable ordinalAn indecomposable ordinal is an ordinal which is not the sum of two smaller ordinals

indescribable cardinalAn indescribable cardinal is a type of large cardinal that cannot be described in a certain language

ineffable cardinalAn ineffable cardinal is a type of large cardinal related to the generalized Kurepa hypothesis whose consistency strength lies betweenthat of subtle cardinals and remarkable cardinals

inner modelAn inner model is a transitive model of ZF containing all ordinals

J

jAn elementary embedding

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JLevels of the Jensen hierarchy

Jensen1. Ronald Jensen2. The Jensen hierarchy is a variation of the constructible hierarchy3. Jensen's covering theorem states that if 0<sup# does not exist then every uncountable set of ordinals is contained in a constructibleset of the same cardinality

Jónsson1. Bjarni Jónsson

2. A Jónsson cardinal is a large cardinal such that for every function f: [κ]<ω → κ there is a set H of order type κ such that for each n, frestricted to n-element subsets of H omits at least one value in κ.3. Jónsson function is a function with the property that, for any subset y of x with the same cardinality as x, the restrictionof to is surjective on .

K

KHKurepa's hypothesis

KPKripke–Platek set theory

Kripke–Platek set theoryKripke–Platek set theory consists roughly of the predicative parts of set theory

Kurepa1. Đuro Kurepa2. The Kurepa hypothesis states that Kurepa trees exist3. A Kurepa tree is a tree (T, <) of height , each of whose levels is at most countable, and has at least many branches

L

L1. L is the constructible universe, and Lα is the hierarchy of constructible sets

2. Lκλ is an infinitary language

large cardinalRoughly, a type of cardinal whose existence cannot be proved in ZFC; see large cardinal

Laver functionA Laver function is a function related to supercompact cardinals that takes ordinals to sets

Lévy1. Azriel Lévy2. The Lévy collapse is a way of destroying cardinals3. The Lévy hierarchy classifies formulas in terms of the number of alternations of unbounded quantifiers

limit

A limit cardinal is a cardinal, usually assumed to be nonzero, that is not the successor κ+ of another cardinal κA limit ordinal is an ordinal, usually assumed to be nonzero, that is not the successor α+1 of another ordinal α

LSTThe language of set theory (with a single binary relation ∈)

M

�

The smallest cardinal at which Martin's axiom fails

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MA model of ZF set theory

MAMartin's axiom

Mac Lane set theoryZermelo set theory with the axiom of separation restricted to formulas with bounded quantifiers

Mahlo cardinalA Mahlo cardinal is an inaccessible cardinal such that the set of inaccessible cardinals less than it is stationary

Martin's axiomMartin's axiom for a cardinal κ states that for any partial order P satisfying the countable chain condition and any family D of dense setsin P of cardinality at most κ, there is a filter F on P such that F ∩ d is non-empty for every d in D

Martin's maximumMartin's maximum states that if D is a collection of dense subsets of a notion of forcing that preserves stationary subsets of ω1, then

there is a D-generic filter

meager

meagreA meager set is one that is the union of a countable number of nowhere-dense sets. Also called a set of first category.

measurable cardinalA measurable cardinal is a cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ.Most (but not all) authors add the condition that it should be uncountable

Milner–Rado paradox

The Milner–Rado paradox states that every ordinal number α less than the successor κ+ of some cardinal number κ can be written as the

union of sets X1,X2,... where Xn is of order type at most κn for n a positive integer.

MMMartin's maximum

morassA morass is a tree with ordinals associated to the nodes and some further structure, satisfying some rather complicated axioms.

Mostowski collapseThe Mostowski collapse is a transitive class associated to a well founded extensional set-like relation.

mouseA certain kind of structure used in constructing core models; see mouse (set theory)

N

N1. The set of natural numbers

2. The Baire space ωω

naive set theory1. Naive set theory can mean set theory developed non-rigorously without axioms2. Naive set theory can mean the inconsistent theory with the axioms of extensionality and comprehension3. Naive set theory is an introductory book on set theory by Halmos

nonstationaryA subset of an ordinal is called nonstationary if it is not stationary, in other words if its complement contains a club setThe nonstationary ideal INS is the ideal of nonstationary sets

normal1. A normal function is a continuous strictly increasing function from ordinals to ordinals

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2. A normal filter or normal measure on an ordinal is a filter or measure closed under diagonal intersections3. The Cantor normal form of an ordinal is its base ω expansion.

NSNonstationary

nullGerman for zero, occasionally used in terms such as "aleph null" (aleph zero) or "null set" (empty set)

number classThe first number class consists of finite ordinals, and the second number class consists of countable ordinals.

O

OCAThe open coloring axiom

ODThe ordinal definable sets

Omega logicΩ-logic is a form of logic introduced by Hugh Woodin

OnThe class of all ordinals

ordinalAn ordinal is the order type of a well-ordered set, usually represented by a von Neumann ordinal, a transitive set well ordered by ∈.

otAbbreviation for "order type of"

P

�

The pseudo-intersection number, the smallest cardinality of a family of infinite subsets of ω that has the strong finite intersectionproperty but has no infinite pseudo-intersection.

P1. The powerset function2. A poset

pairing functionA pairing function is a bijection from X×X to X for some set X

pantachie

pantachyA pantachy is a maximal chain of a poset

paradox1. Burali-Forti paradox2. Cantor's paradox3. Hilbert's paradox4. Russell's paradox

partial order1. A set with a transitive antisymmetric relation2. A set with a transitive symmetric relation

partition cardinalAn alternative name for an Erdős cardinal

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PCFAbbreviation for "possible cofinalities", used in PCF theory

PDThe axiom of projective determinacy

perfect setA perfect set is a subset of a topological set equal to its derived set

permutation modelA permutation model of ZFA is constructed using a group

PFAThe proper forcing axiom

po

posetA set with a partial order

Polish spaceA Polish space is a separable topological space homeomorphic to a complete metric space

power1. "Power" is an archaic term for cardinality2. The powerset or power set of a set is the set of all its subsets

projective1. A projective set is a set that can be obtained from an analytic set by repeatedly taking complements and projections2. Projective determinacy is an axiom asserting that projective sets are determined

proper1. A proper class is a class that is not a set2. A proper subset of a set X is a subset not equal to X.3. A proper forcing is a forcing notion that does not collapse any stationary set4. The proper forcing axiom asserts that if P is proper and Dα is a dense subset of P for each α<ω1, then there is a filter G P such that

Dα ∩ G is nonempty for all α<ω1

Q

QThe (ordered set of) rational numbers

R

R1. Rα is an alternative name for the level Vα of the von Neumann hierarchy.

2. The real numbers

Ramsey cardinalA Ramsey cardinal is a large cardinal satisfying a certain partition condition

ranThe range of a function

rank1. The rank of a set is the smallest ordinal greater than the ranks of its elements2. A rank Vα is the collection of all sets of rank less than α, for an ordinal α

3. rank-into-rank is a type of large cardinal (axiom)

reflecting cardinalA reflecting cardinal is a type of large cardinal whose strength lies between being weakly compact and Mahlo

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reflection principleA reflection principle states that there is a set similar in some way to the universe of all sets

regressiveA function f from a subset of an ordinal to the ordinal is called regressive if f(α)<α for all α in its domain

regularA regular cardinal is one equal to its own cofinality.

Reinhardt cardinalA Reinhardt cardinal is a cardinal in a model V of ZF that is the critical point of an elementary embedding of V into itself

relationA set or class whose elements are ordered pairs

ROThe regular open sets of a topological space or poset

Rowbottom cardinalA Rowbottom cardinal is a large cardinal satisfying a certain partition condition

rudThe rudimentary closure of a set

rudimentaryA rudimentary function is a functions definable by certain elementary operations, used in the construction of the Jensen hierarchy

Russell's paradoxRussell's paradox is that the set of all sets not containing themselves is contradictory so cannot exist

S

SCHSingular cardinal hypothesis

Scott's trickScott's trick is a way of coding proper equivalence classes by sets by taking the elements of the class of smallest rank

second1. A set of second category is a set that is not of first category: in other words a set that is not the union of a countable number ofnowhere-dense sets.2. An ordinal of the second class is a countable infinite ordinal3. Second order logic allows quantification over subsets as well as over elements of a model

separating set1. A separating set is a set containing a given set and disjoint from another given set2. A separating set is a set S of functions on a set such that for any two distinct points there is a function in S with different values onthem.

separativeA separative poset is one that can be densely embedded into the poset of nonzero elements of a Boolean algebra.

SFIPStrong finite intersection property

SHSuslin's hypothesis

Shelah cardinalA Shelah cardinal is a large cardinal that is the critical point of an elementary embedding satisfying certain conditions

shrewd cardinalA shrewd cardinal is a type of large cardinal generalizing indecribable cardinals to transfinite levels

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Sierpinski

Sierpiński1. Wacław Sierpiński2. A Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable

singular1. A singular cardinal is one that is not regular

2. The singular cardinal hypothesis states that if κ is any singular strong limit cardinal, then 2κ = κ+.

SOCASemi open coloring axiom

Solovay modelThe Solovay model is a model of ZF in which every set of reals is measurable

specialA special Aronszajn tree is one with an order preserving map to the rationals

squareThe square principle is a combinatorial principle holding in the constructible universe and some other inner models

standard modelA model of set theory where the relation ∈ is the same as the usual one.

stationary setA stationary set is a subset of an ordinal intersecting every club set

strong1. The strong finite intersection property says that the intersection of any finite number of elements of a set is infinite2. A strong cardinal is a cardinal κ such that if λ is any ordinal, there is an elementary embedding with critical point κ from the universeinto a transitive inner model containing all elements of Vλ

3. A strong limit cardinal is a (usually nonzero) cardinal that is larger then the powerset of any smaller cardinal

strongly compact cardinalA strongly compact cardinal is a cardinal κ such that every κ-complete filter can be extended to a κ complete ultrafilter

strongly inaccessible cardinalA strongly inaccessible cardinal is a regular stong limit cardinal

subtle cardinalA subtle cardinal is a type of large cardinal closely related to ethereal cardinals

successor1. A successor cardinal is the smallest cardinal larger than some given cardinal1. A successor ordinal is the smallest ordinal larger than some given ordinal

sunflowerA sunflower, also called a delta system, is a collection of sets such that any two distinct sets have intersection X for some fixed set X

Souslin

Suslin0. Mikhail Yakovlevich Suslin (sometimes written Souslin)1. A Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition

2. A Suslin cardinal is a cardinal λ such that there exists a set P ⊂ 2ω such that P is λ-Suslin but P is not λ'-Suslin for any λ' < λ.3. The Suslin hypothesis says that Suslin lines do not exist4. A Suslin line is a complete dense unbounded totally ordered set satisfying the countable chain condition5. The Suslin number is the supremum of the cardinalities of families of disjoint open non-empty sets6. The Suslin operation, usually denoted by A, is an operation that constructs a set from a Suslin scheme7. The Suslin problem asks whether Suslin lines exist8. The Suslin property states that there is no uncountable family of pairwise disjoint non-empty open subsets

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9. A Suslin representation of a set of reals is a tree whose projection is that set of reals10. A Suslin scheme is a function with domain the finite sequences of positive integers11. A Suslin set is a set that is the image of a tree under a certain projection12. A Suslin space is the image of a Polish space under a continuous mapping13. A Suslin subset is a subset that is the image of a tree under a certain projection14. The Suslin theorem about analytic sets states that a set that is analytic and coanalytic is Borel15. A Suslin tree is a tree of height ω1 such that every branch and every antichain is at most countable.

super transitive

supertransitiveA supertransitive set is a set that contains all subsets of all its elements

symmetric modelA symmetric model is a model of ZF (without the axiom of choice) constructed using a group action on a forcing poset

T

TA tree

tall cardinalA tall cardinal is a type of large cardinal that is the critical point of a certain sort of elementary embedding

Tarski's theorem1. Tarski's theorem that the axiom of choice is equivalent to the existence of a bijection from X to X×X for all sets X

TCThe transitive closure of a set

total orderA total order is a relation that is transitive and antisymmetric such that any two elements are comparable

transitive1. A transitive relation2. The transitive closure of a set is the smallest transitive set containing it.3. A transitive set or class is a set or class such that the membership relation is transitive on it.4. A transitive model is a model of set theory that is transitive and has the usual membership relation

tree1. A tree is a partially ordered set (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation <2. A tree is a collection of finite sequences such that every prefix of a sequence in the collection also belongs to the collection.3. A cardinal κ has the tree property if there are no κ-Aronszajn trees

U

Ulam matrixAn Ulam matrix is a collection of subsets of a cardinal indexed by pairs of ordinals, that satisfies certain properties.

UltAn ultrapower or ultraproduct

ultrapowerAn ultraproduct in which all factors are equal

ultraproductAn ultraproduct is the quotient of a product of models by a certain equivalence relation

unfoldable cardinalAn unfoldable cardinal a cardinal κ such that for every ordinal λ and every transitive model M of cardinality κ of ZFC-minus-power setsuch that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into atransitive model with the critical point of j being κ and j(κ) ≥ λ.

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uniformizationUniformization is a weak form of the axiom of choice, giving cross sections for special subsets of a product of two Polish spaces

universal

universeThe universal class, or universe, is the class of all sets.

urelementAn urelement is something that is not a set but allowed to be an element of a set

V

VV is the universe of all sets, and the sets Vα form the Von Neumann hierarchy

V=LThe axiom of constructibility

VeblenThe Veblen hierarchy is a family of ordinal valued functions, special cases of which are called Veblen functions.

von Neumann1. John von Neumann2. A von Neumann ordinal is an ordinal encoded as the union of all smaller (von Neumann) ordinals3. The von Neumann hierarchy is a cumulative hierarchy Vα with Vα+1 the powerset of Vα.

Vopenka

Vopěnka1. Petr Vopěnka2. Vopěnka's principle states that for every proper class of binary relations there is one elementarily embeddable into another3. A Vopěnka cardinal is an inaccessible cardinal κ such that and Vopěnka's principle holds for Vκ

W

weakly inaccessible cardinalA weakly inaccessible cardinal is a regular weak limit cardinal

weakly compact cardinalA weakly compact cardinal is a cardinal κ (usually also assumed to be inaccessible) such that the infinitary language Lκ,κ satisfies the

weak compactness theorem

well foundedA relation is called well founded if every non-empty subset has a minimal element

well orderingA well ordering is a well founded relation, usually also assumed to be a total order

Woodin cardinalA Woodin cardinal is a type of large cardinal that is the critial point of a certain sort of elementary embedding, closely related to theaxiom of projective determinacy

XYZ

ZZermelo set theory without the axiom of choice

ZCZermelo set theory with the axiom of choice

Zermelo1. Ernst Zermelo

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2. Zermelo−Fraenkel set theory is the standard system of axioms for set theory3. Zermelo set theory is similar to the usual Zermelo-Fraenkel set theory, but without the axioms of replacement and foundation4. Zermelo's well-ordering theorem states that every set can be well ordered

ZFZermelo−Fraenkel set theory without the axiom of choice

ZFAZermelo−Fraenkel set theory with atoms

ZFCZermelo−Fraenkel set theory with the axiom of choice

See also

Glossary of Principia Mathematica

References

Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag.ISBN 978-3-540-44085-7. Zbl 1007.03002.

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