Ring of sets 1_8

Ring of sets 1From Wikipedia, the free encyclopedia

Contents

1 Birkhoffs representation theorem 11.1 Understanding the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The partial order of join-irreducibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Birkhoffs theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Rings of sets and preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Borel set 72.1 Generating the Borel algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Standard Borel spaces and Kuratowski theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Non-Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Alternative non-equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Cyclic order 113.1 Finite cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 The ternary relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Rolling and cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.3 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.4 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Monotone functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.1 Functions on finite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Further constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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3.4.1 Unrolling and covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.2 Products and retracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.6.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6.2 Modified axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.7 Symmetries and model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.9 Notes on usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Delta-ring 254.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Distributive lattice 265.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4 Characteristic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.5 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.6 Free distributive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Dynkin system 326.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Empty set 347.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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7.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8 Family of sets 408.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9 Field of sets 429.1 Fields of sets in the representation theory of Boolean algebras . . . . . . . . . . . . . . . . . . . . 42

9.1.1 Stone representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.1.2 Separative and compact fields of sets: towards Stone duality . . . . . . . . . . . . . . . . . 42

9.2 Fields of sets with additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.2.1 Sigma algebras and measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.2.2 Topological fields of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.2.3 Preorder fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.2.4 Complex algebras and fields of sets on relational structures . . . . . . . . . . . . . . . . . . 44

9.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10 Ideal (set theory) 4610.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2 Examples of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.2.1 General examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2.2 Ideals on the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2.3 Ideals on the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.2.4 Ideals on other sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

10.3 Operations on ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.4 Relationships among ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

11 Intersection (set theory) 4911.1 Basic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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11.1.1 Intersecting and disjoint sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.2 Arbitrary intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.3 Nullary intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

12 Lebesgue measure 5512.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

12.1.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5612.4 Null sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.5 Construction of the Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.6 Relation to other measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

13 Mathematics 6013.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

13.1.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6113.1.2 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

13.2 Definitions of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.2.1 Mathematics as science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

13.3 Inspiration, pure and applied mathematics, and aesthetics . . . . . . . . . . . . . . . . . . . . . . . 6713.4 Notation, language, and rigor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.5 Fields of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

13.5.1 Foundations and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.5.2 Pure mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.5.3 Applied mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

13.6 Mathematical awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

14 Measure (mathematics) 7814.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7814.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7914.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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14.3.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8014.3.2 Measures of infinite unions of measurable sets . . . . . . . . . . . . . . . . . . . . . . . . 8014.3.3 Measures of infinite intersections of measurable sets . . . . . . . . . . . . . . . . . . . . . 80

14.4 Sigma-finite measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8014.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8114.6 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8114.7 Non-measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8114.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8114.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8214.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8214.11Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8314.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

15 Non-measurable set 8615.1 Historical constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.3 Consistent definitions of measure and probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 8715.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8715.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

15.5.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8715.5.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

16 Pi system 8816.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8816.2 Relationship to -Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

16.2.1 The - Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8916.3 -Systems in Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

16.3.1 Equality in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9016.3.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

16.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9116.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9116.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

17 Ring of sets 9217.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9217.2 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9317.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9317.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

18 Semiring 9418.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9418.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

18.2.1 In general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

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18.2.2 Specific examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9518.3 Semiring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9618.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9618.5 Complete and continuous semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9618.6 Star semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

18.6.1 Complete star semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9718.7 Further generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9818.8 Semiring of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9818.9 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9818.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9818.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9818.12Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9818.13Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

19 Set (mathematics) 10119.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10219.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10219.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

19.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10419.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

19.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10519.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10519.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

19.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10619.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10719.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10719.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

19.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11019.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11019.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11119.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11119.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11219.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11219.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11219.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

20 Sigma-algebra 11320.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

20.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11320.1.2 Limits of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11420.1.3 Sub -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

20.2 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

CONTENTS vii

20.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11520.2.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11520.2.3 Combining -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11520.2.4 -algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11620.2.5 Relation to -ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11620.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

20.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11720.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11720.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

20.4 -algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11720.4.1 -algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 11720.4.2 -algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11720.4.3 Borel and Lebesgue -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11820.4.4 Product -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11820.4.5 -algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11820.4.6 -algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 11920.4.7 -algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 119

20.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11920.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12020.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

21 Sigma-ideal 12121.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

22 Sigma-ring 12222.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12222.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12222.3 Similar concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12222.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12222.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12322.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

23 Subset 12423.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12523.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12523.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12523.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12623.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12623.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12623.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

24 Unit interval 12824.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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24.1.1 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12824.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12924.3 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12924.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12924.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

25 Upper set 13025.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13125.2 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13125.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13125.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13125.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 132

25.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13225.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13625.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Chapter 1

Birkhoffs representation theorem

This is about lattice theory. For other similarly named results, see Birkhoffs theorem (disambiguation).

In mathematics, Birkhoffs representation theorem for distributive lattices states that the elements of any finitedistributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions andintersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributivelattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spacesand preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.[1]

The name Birkhoffs representation theorem has also been applied to two other results of Birkhoff, one from 1935on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and BirkhoffsHSP theorem representing algebras as products of irreducible algebras. Birkhoffs representation theorem has alsobeen called the fundamental theorem for finite distributive lattices.[2]

1.1 Understanding the theorem

Many lattices can be defined in such a way that the elements of the lattice are represented by sets, the join operationof the lattice is represented by set union, and the meet operation of the lattice is represented by set intersection. Forinstance, the Boolean lattice defined from the family of all subsets of a finite set has this property. More generally anyfinite topological space has a lattice of sets as its family of open sets. Because set unions and intersections obey thedistributive law, any lattice defined in this way is a distributive lattice. Birkhoffs theorem states that in fact all finitedistributive lattices can be obtained this way, and later generalizations of Birkhoffs theorem state a similar thing forinfinite distributive lattices.

1.2 Examples

Consider the divisors of some composite number, such as (in the figure) 120, partially ordered by divisibility. Anytwo divisors of 120, such as 12 and 20, have a unique greatest common factor 12 20 = 4, the largest number thatdivides both of them, and a unique least common multiple 12 20 = 60; both of these numbers are also divisors of120. These two operations and satisfy the distributive law, in either of two equivalent forms: (x y) z = (x z) (y z) and (x y) z = (x z) (y z), for all x, y, and z. Therefore, the divisors form a finite distributive lattice.One may associate each divisor with the set of prime powers that divide it: thus, 12 is associated with the set {2,3,4},while 20 is associated with the set {2,4,5}. Then 12 20 = 4 is associated with the set {2,3,4} {2,4,5} = {2,4},while 12 20 = 60 is associated with the set {2,3,4} {2,4,5} = {2,3,4,5}, so the join and meet operations of thelattice correspond to union and intersection of sets.The prime powers 2, 3, 4, 5, and 8 appearing as elements in these sets may themselves be partially ordered bydivisibility; in this smaller partial order, 2 4 8 and there are no order relations between other pairs. The 16 setsthat are associated with divisors of 120 are the lower sets of this smaller partial order, subsets of elements such thatif x y and y belongs to the subset, then x must also belong to the subset. From any lower set L, one can recover the

1
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2 CHAPTER 1. BIRKHOFFS REPRESENTATION THEOREM

{2} {3} {5}

{2,4} {2,3} {2,5} {3,5}

{2,4,8} {2,3,4} {2,4,5} {2,3,5}

{2,3,4,8} {2,4,5,8} {2,3,4,5}

{2,3,4,5,8}

1

2 3 5

4 6 10 15

8 12 20 30

24 40 60

120

The distributive lattice of divisors of 120, and its representation as sets of prime powers.

associated divisor by computing the least common multiple of the prime powers in L. Thus, the partial order on thefive prime powers 2, 3, 4, 5, and 8 carries enough information to recover the entire original 16-element divisibilitylattice.Birkhoffs theorem states that this relation between the operations and of the lattice of divisors and the operations and of the associated sets of prime powers is not coincidental, and not dependent on the specific properties ofprime numbers and divisibility: the elements of any finite distributive lattice may be associated with lower sets of apartial order in the same way.As another example, the application of Birkhoffs theorem to the family of subsets of an n-element set, partiallyordered by inclusion, produces the free distributive lattice with n generators. The number of elements in this latticeis given by the Dedekind numbers.

1.3 The partial order of join-irreducibles

In a lattice, an element x is join-irreducible if x is not the join of a finite set of other elements. Equivalently, x isjoin-irreducible if it is neither the bottom element of the lattice (the join of zero elements) nor the join of any twosmaller elements. For instance, in the lattice of divisors of 120, there is no pair of elements whose join is 4, so 4 isjoin-irreducible. An element x is join-prime if, whenever x y z, either x y or x z. In the same lattice, 4 isjoin-prime: whenever lcm(y,z) is divisible by 4, at least one of y and z must itself be divisible by 4.In any lattice, a join-prime element must be join-irreducible. Equivalently, an element that is not join-irreducible isnot join-prime. For, if an element x is not join-irreducible, there exist smaller y and z such that x = y z. But then x y z, and x is not less than or equal to either y or z, showing that it is not join-prime.There exist lattices in which the join-prime elements form a proper subset of the join-irreducible elements, but ina distributive lattice the two types of elements coincide. For, suppose that x is join-irreducible, and that x y z.This inequality is equivalent to the statement that x = x (y z), and by the distributive law x = (x y) (x z).But since x is join-irreducible, at least one of the two terms in this join must be x itself, showing that either x = x y(equivalently x y) or x = x z (equivalently x z).The lattice ordering on the subset of join-irreducible elements forms a partial order; Birkhoffs theorem states thatthe lattice itself can be recovered from the lower sets of this partial order.

1.4 Birkhoffs theorem

In any partial order, the lower sets form a lattice in which the lattices partial ordering is given by set inclusion, thejoin operation corresponds to set union, and the meet operation corresponds to set intersection, because unions andintersections preserve the property of being a lower set. Because set unions and intersections obey the distributive
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1.5. RINGS OF SETS AND PREORDERS 3

law, this is a distributive lattice. Birkhoffs theorem states that any finite distributive lattice can be constructed in thisway.

Theorem. Any finite distributive lattice L is isomorphic to the lattice of lower sets of the partial orderof the join-irreducible elements of L.

That is, there is a one-to-one order-preserving correspondence between elements of L and lower sets of the partialorder. The lower set corresponding to an element x of L is simply the set of join-irreducible elements of L that areless than or equal to x, and the element of L corresponding to a lower set S of join-irreducible elements is the join ofS.If one starts with a lower set S of join-irreducible elements, lets x be the join of S, and constructs lower set T of thejoin-irreducible elements less than or equal to x, then S = T. For, every element of S clearly belongs to T, and anyjoin-irreducible element less than or equal to x must (by join-primality) be less than or equal to one of the membersof S, and therefore must (by the assumption that S is a lower set) belong to S itself. Conversely, if one starts with anelement x of L, lets S be the join-irreducible elements less than or equal to x, and constructs y as the join of S, thenx = y. For, as a join of elements less than or equal to x, y can be no greater than x itself, but if x is join-irreduciblethen x belongs to S while if x is the join of two or more join-irreducible items then they must again belong to S, so y x. Therefore, the correspondence is one-to-one and the theorem is proved.

1.5 Rings of sets and preorders

Birkhoff (1937) defined a ring of sets to be a family of sets that is closed under the operations of set unions and setintersections; later, motivated by applications in mathematical psychology, Doignon & Falmagne (1999) called thesame structure a quasi-ordinal knowledge space. If the sets in a ring of sets are ordered by inclusion, they form adistributive lattice. The elements of the sets may be given a preorder in which x y whenever some set in the ringcontains x but not y. The ring of sets itself is then the family of lower sets of this preorder, and any preorder givesrise to a ring of sets in this way.

1.6 Functoriality

Birkhoffs theorem, as stated above, is a correspondence between individual partial orders and distributive lattices.However, it can also be extended to a correspondence between order-preserving functions of partial orders andbounded homomorphisms of the corresponding distributive lattices. The direction of these maps is reversed in thiscorrespondence.Let 2 denote the partial order on the two-element set {0, 1}, with the order relation 0 < 1, and (following Stanley)let J(P) denote the distributive lattice of lower sets of a finite partial order P. Then the elements of J(P) correspondone-for-one to the order-preserving functions from P to 2.[2] For, if is such a function, 1(0) forms a lower set,and conversely if L is a lower set one may define an order-preserving function L that maps L to 0 and that mapsthe remaining elements of P to 1. If g is any order-preserving function from Q to P, one may define a function g*from J(P) to J(Q) that uses the composition of functions to map any element L of J(P) to L g. This compositefunction maps Q to 2 and therefore corresponds to an element g*(L) = (L g)1(0) of J(Q). Further, for any x andy in J(P), g*(x y) = g*(x) g*(y) (an element of Q is mapped by g to the lower set x y if and only if belongsboth to the set of elements mapped to x and the set of elements mapped to y) and symmetrically g*(x y) = g*(x) g*(y). Additionally, the bottom element of J(P) (the function that maps all elements of P to 0) is mapped by g* tothe bottom element of J(Q), and the top element of J(P) is mapped by g* to the top element of J(Q). That is, g* is ahomomorphism of bounded lattices.However, the elements of P themselves correspond one-for-one with bounded lattice homomorphisms from J(P) to 2.For, if x is any element of P, one may define a bounded lattice homomorphism jx that maps all lower sets containingx to 1 and all other lower sets to 0. And, for any lattice homomorphism from J(P) to 2, the elements of J(P) thatare mapped to 1 must have a unique minimal element x (the meet of all elements mapped to 1), which must bejoin-irreducible (it cannot be the join of any set of elements mapped to 0), so every lattice homomorphism has theform jx for some x. Again, from any bounded lattice homomorphism h from J(P) to J(Q) one may use composition offunctions to define an order-preserving map h* from Q to P. It may be verified that g** = g for any order-preservingmap g from Q to P and that and h** = h for any bounded lattice homomorphism h from J(P) to J(Q).
https://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theorem#CITEREFBirkhoff1937https://en.wikipedia.org/wiki/Family_of_setshttps://en.wikipedia.org/wiki/Closure_(mathematics)https://en.wikipedia.org/wiki/Mathematical_psychologyhttps://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theorem#CITEREFDoignonFalmagne1999https://en.wikipedia.org/wiki/Knowledge_spacehttps://en.wikipedia.org/wiki/Preorderhttps://en.wikipedia.org/wiki/Lattice_homomorphismhttps://en.wikipedia.org/wiki/Composition_of_functions


In category theoretic terminology, J is a contravariant hom-functor J = Hom(,2) that defines a duality of categoriesbetween, on the one hand, the category of finite partial orders and order-preserving maps, and on the other hand thecategory of finite distributive lattices and bounded lattice homomorphisms.

1.7 Generalizations

In an infinite distributive lattice, it may not be the case that the lower sets of the join-irreducible elements are inone-to-one correspondence with lattice elements. Indeed, there may be no join-irreducibles at all. This happens, forinstance, in the lattice of all natural numbers, ordered with the reverse of the usual divisibility ordering (so x ywhen y divides x): any number x can be expressed as the join of numbers xp and xq where p and q are distinct primenumbers. However, elements in infinite distributive lattices may still be represented as sets via Stones representationtheorem for distributive lattices, a form of Stone duality in which each lattice element corresponds to a compact openset in a certain topological space. This generalized representation theorem can be expressed as a category-theoreticduality between distributive lattices and coherent spaces (sometimes called spectral spaces), topological spaces inwhich the compact open sets are closed under intersection and form a base for the topology.[3] Hilary Priestley showedthat Stones representation theorem could be interpreted as an extension of the idea of representing lattice elementsby lower sets of a partial order, using Nachbins idea of ordered topological spaces. Stone spaces with an additionalpartial order linked with the topology via Priestley separation axiom can also be used to represent bounded distributivelattices. Such spaces are known as Priestley spaces. Further, certain bitopological spaces, namely pairwise Stonespaces, generalize Stones original approach by utilizing two topologies on a set to represent an abstract distributvelattice. Thus, Birkhoffs representation theorem extends to the case of infinite (bounded) distributive lattices in atleast three different ways, summed up in duality theory for distributive lattices.Birkhoffs representation theorem may also be generalized to finite structures other than distributive lattices. In adistributive lattice, the self-dual median operation[4]

m(x, y, z) = (x y) (x z) (y z) = (x y) (x z) (y z)

gives rise to a median algebra, and the covering relation of the lattice forms a median graph. Finite median algebrasand median graphs have a dual structure as the set of solutions of a 2-satisfiability instance; Barthlemy & Constantin(1993) formulate this structure equivalently as the family of initial stable sets in a mixed graph.[5] For a distributivelattice, the corresponding mixed graph has no undirected edges, and the initial stable sets are just the lower sets ofthe transitive closure of the graph. Equivalently, for a distributive lattice, the implication graph of the 2-satisfiabilityinstance can be partitioned into two connected components, one on the positive variables of the instance and theother on the negative variables; the transitive closure of the positive component is the underlying partial order of thedistributive lattice.Another result analogous to Birkhoffs representation theorem, but applying to a broader class of lattices, is thetheorem of Edelman (1980) that any finite join-distributive lattice may be represented as an antimatroid, a familyof sets closed under unions but in which closure under intersections has been replaced by the property that eachnonempty set has a removable element.

1.8 Notes

[1] Birkhoff (1937).

[2] (Stanley 1997).

[3] Johnstone (1982).

[4] Birkhoff & Kiss (1947).

[5] A minor difference between the 2-SAT and initial stable set formulations is that the latter presupposes the choice of a fixedbase point from the median graph that corresponds to the empty initial stable set.
https://en.wikipedia.org/wiki/Category_theoryhttps://en.wikipedia.org/wiki/Hom-functorhttps://en.wikipedia.org/wiki/Equivalence_of_categorieshttps://en.wikipedia.org/wiki/Prime_numberhttps://en.wikipedia.org/wiki/Prime_numberhttps://en.wikipedia.org/wiki/Stone%2527s_representation_theoremhttps://en.wikipedia.org/wiki/Stone%2527s_representation_theoremhttps://en.wikipedia.org/wiki/Stone_dualityhttps://en.wikipedia.org/wiki/Compact_spacehttps://en.wikipedia.org/wiki/Open_sethttps://en.wikipedia.org/wiki/Open_sethttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Category_theoryhttps://en.wikipedia.org/wiki/Equivalence_of_categorieshttps://en.wikipedia.org/wiki/Coherent_spacehttps://en.wikipedia.org/wiki/Spectral_spacehttps://en.wikipedia.org/wiki/Base_(topology)https://en.wikipedia.org/wiki/Hilary_Priestleyhttps://en.wikipedia.org/wiki/Priestley_spacehttps://en.wikipedia.org/wiki/Priestley_spacehttps://en.wikipedia.org/wiki/Bitopological_spacehttps://en.wikipedia.org/wiki/Pairwise_Stone_spacehttps://en.wikipedia.org/wiki/Pairwise_Stone_spacehttps://en.wikipedia.org/wiki/Duality_theory_for_distributive_latticeshttps://en.wikipedia.org/wiki/Median_algebrahttps://en.wikipedia.org/wiki/Median_graphhttps://en.wikipedia.org/wiki/2-satisfiabilityhttps://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theorem#CITEREFBarth.C3.A9lemyConstantin1993https://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theorem#CITEREFBarth.C3.A9lemyConstantin1993https://en.wikipedia.org/wiki/Independent_set_(graph_theory)https://en.wikipedia.org/wiki/Mixed_graphhttps://en.wikipedia.org/wiki/Transitive_closurehttps://en.wikipedia.org/wiki/Implication_graphhttps://en.wikipedia.org/wiki/Connected_component_(graph_theory)https://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theorem#CITEREFEdelman1980https://en.wikipedia.org/wiki/Antimatroidhttps://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theorem#CITEREFBirkhoff1937https://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theorem#CITEREFStanley1997https://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theorem#CITEREFJohnstone1982https://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theorem#CITEREFBirkhoffKiss1947

1.9. REFERENCES 5

1.9 References Barthlemy, J.-P.; Constantin, J. (1993), Median graphs, parallelism and posets, Discrete Mathematics 111

(13): 4963, doi:10.1016/0012-365X(93)90140-O.

Birkhoff, Garrett (1937), Rings of sets, Duke Mathematical Journal 3 (3): 443454, doi:10.1215/S0012-7094-37-00334-X.

Birkhoff, Garrett; Kiss, S. A. (1947), A ternary operation in distributive lattices, Bulletin of the AmericanMathematical Society 53 (1): 749752, doi:10.1090/S0002-9904-1947-08864-9, MR 0021540.

Doignon, J.-P.; Falmagne, J.-Cl. (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2.

Edelman, Paul H. (1980), Meet-distributive lattices and the anti-exchange closure, Algebra Universalis 10(1): 290299, doi:10.1007/BF02482912.

Johnstone, Peter (1982), II.3 Coherent locales, Stone Spaces, Cambridge University Press, pp. 6269, ISBN978-0-521-33779-3.

Priestley, H. A. (1970), Representation of distributive lattices by means of ordered Stone spaces, Bulletin ofthe London Mathematical Society 2 (2): 186190, doi:10.1112/blms/2.2.186.

Priestley, H. A. (1972), Ordered topological spaces and the representation of distributive lattices, Proceedingsof the London Mathematical Society 24 (3): 507530, doi:10.1112/plms/s3-24.3.507.

Stanley, R. P. (1997), Enumerative Combinatorics, Volume I, Cambridge Studies in Advanced Mathematics 49,Cambridge University Press, pp. 104112.
https://en.wikipedia.org/wiki/Discrete_Mathematics_(journal)https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1016%252F0012-365X%252893%252990140-Ohttps://en.wikipedia.org/wiki/Garrett_Birkhoffhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1215%252FS0012-7094-37-00334-Xhttps://dx.doi.org/10.1215%252FS0012-7094-37-00334-Xhttps://en.wikipedia.org/wiki/Garrett_Birkhoffhttp://projecteuclid.org/euclid.bams/1183510977https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1090%252FS0002-9904-1947-08864-9https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=0021540https://en.wikipedia.org/wiki/Jean-Claude_Falmagnehttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/3-540-64501-2https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252FBF02482912https://en.wikipedia.org/wiki/Peter_Johnstone_(mathematician)https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-33779-3https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1112%252Fblms%252F2.2.186https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1112%252Fplms%252Fs3-24.3.507https://en.wikipedia.org/wiki/Richard_P._Stanley


Distributive example lattice, with join-irreducible elements a,...,g (shadowed nodes). The lower set a node corresponds to by Birkhoffsisomorphism is shown in light blue.

Chapter 2

Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, fromclosed sets) through the operations of countable union, countable intersection, and relative complement. Borel setsare named after mile Borel.For a topological space X, the collection of all Borel sets on X forms a -algebra, known as the Borel algebra orBorel -algebra. The Borel algebra on X is the smallest -algebra containing all open sets (or, equivalently, all closedsets).Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closedsets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called aBorel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather thanthe open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff -compactspaces, but can be different in more pathological spaces.

2.1 Generating the Borel algebra

In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows.For a collection T of subsets of X (that is, for any subset of the power set P(X) of X), let

T be all countable unions of elements of T

T be all countable intersections of elements of T

T = (T).

Now define by transfinite induction a sequence Gm, where m is an ordinal number, in the following manner:

For the base case of the definition, let G0 be the collection of open subsets of X.

If i is not a limit ordinal, then i has an immediately preceding ordinal i 1. Let

Gi = [Gi1].

If i is a limit ordinal, set

Gi =j

8 CHAPTER 2. BOREL SET

G 7 G.

to the first uncountable ordinal.To prove this claim, note that any open set in a metric space is the union of an increasing sequence of closed sets.In particular, complementation of sets maps Gm into itself for any limit ordinal m; moreover if m is an uncountablelimit ordinal, Gm is closed under countable unions.Note that for each Borel set B, there is some countable ordinal B such that B can be obtained by iterating theoperation over B. However, as B varies over all Borel sets, B will vary over all the countable ordinals, and thus thefirst ordinal at which all the Borel sets are obtained is 1, the first uncountable ordinal.

2.1.1 Example

An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers. It isthe algebra on which the Borel measure is defined. Given a real random variable defined on a probability space, itsprobability distribution is by definition also a measure on the Borel algebra.The Borel algebra on the reals is the smallest -algebra on R which contains all the intervals.In the construction by transfinite induction, it can be shown that, in each step, the number of sets is, at most, thepower of the continuum. So, the total number of Borel sets is less than or equal to

1 20 = 20 .

2.2 Standard Borel spaces and Kuratowski theorems

Let X be a topological space. The Borel space associated to X is the pair (X,B), where B is the -algebra of Borelsets of X.Mackey defined a Borel space somewhat differently, writing that it is a set together with a distinguished -field ofsubsets called its Borel sets. [1] However, modern usage is to call the distinguished sub-algebra measurable sets andsuch spaces measurable spaces. The reason for this distinction is that the Borel sets are the -algebra generated byopen sets (of a topological space), whereas Mackeys definition refers to a set equipped with an arbitrary -algebra.There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.[2]

Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. Afunction f : X Y is measurable if it pulls back measurable sets, i.e., for all measurable sets B in Y, f1(B) is ameasurable set in X.Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X which defines thetopology of X and which makes X a complete separable metric space. Then X as a Borel space is isomorphic to oneof (1) R, (2) Z or (3) a finite space. (This result is reminiscent of Maharams theorem.)Considered as Borel spaces, the real line R, the union of R with a countable set, and Rn are isomorphic.A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized upto isomorphism by its cardinality,[3] and any uncountable standard Borel space has the cardinality of the continuum.For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injectivemaps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel.See analytic set.Every probability measure on a standard Borel space turns it into a standard probability space.

2.3 Non-Borel sets

An example of a subset of the reals which is non-Borel, due to Lusin[4] (see Sect. 62, pages 7678), is describedbelow. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved.
https://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Borel_measurehttps://en.wikipedia.org/wiki/Probability_spacehttps://en.wikipedia.org/wiki/Probability_distributionhttps://en.wikipedia.org/wiki/Interval_(mathematics)https://en.wikipedia.org/wiki/Cardinalityhttps://en.wikipedia.org/wiki/Power_of_the_continuumhttps://en.wikipedia.org/wiki/George_Mackeyhttps://en.wikipedia.org/wiki/Measurable_spacehttps://en.wikipedia.org/wiki/Category_(mathematics)https://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Measurable_functionhttps://en.wikipedia.org/wiki/Measurable_functionhttps://en.wikipedia.org/wiki/Pullbackhttps://en.wikipedia.org/wiki/Polish_spacehttps://en.wikipedia.org/wiki/Metric_(mathematics)https://en.wikipedia.org/wiki/Separable_spacehttps://en.wikipedia.org/wiki/Isomorphichttps://en.wikipedia.org/wiki/Maharam%2527s_theoremhttps://en.wikipedia.org/wiki/Polish_spacehttps://en.wikipedia.org/wiki/Analytic_sethttps://en.wikipedia.org/wiki/Probability_measurehttps://en.wikipedia.org/wiki/Standard_probability_spacehttps://en.wikipedia.org/wiki/Nikolai_Luzinhttps://en.wikipedia.org/wiki/Non-measurable_set

2.4. ALTERNATIVE NON-EQUIVALENT DEFINITIONS 9

Every irrational number has a unique representation by a continued fraction

x = a0 +1

a1 +1

a2 +1

a3 +1

. . .

where a0 is some integer and all the other numbers ak are positive integers. Let A be the set of all irrationalnumbers that correspond to sequences (a0, a1, . . . ) with the following property: there exists an infinite subsequence(ak0 , ak1 , . . . ) such that each element is a divisor of the next element. This set A is not Borel. In fact, it is analytic,and complete in the class of analytic sets. For more details see descriptive set theory and the book by Kechris,especially Exercise (27.2) on page 209, Definition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.Another non-Borel set is an inverse image f1[0] of an infinite parity function f : {0, 1} {0, 1} . However, thisis a proof of existence (via the axiom of choice), not an explicit example.

2.4 Alternative non-equivalent definitions

According to Halmos (Halmos 1950, page 219), a subset of a locally compact Hausdorff topological space is calleda Borel set if it belongs to the smallest ring containing all compact sets.Norberg and Vervaat [5] redefine the Borel algebra of a topological space X as the algebra generated by its opensubsets and its compact saturated subsets. This definition is well-suited for applications in the case where X is notHausdorff. It coincides with the usual definition if X is second countable or if every compact saturated subset isclosed (which is the case in particular if X is Hausdorff).

2.5 See also

Baire set

Cylindrical -algebra

Polish space

Descriptive set theory

Borel hierarchy

2.6 References

William Arveson, An Invitation to C*-algebras, Springer-Verlag, 1981. (See Chapter 3 for an excellent expo-sition of Polish topology)

Richard Dudley, Real Analysis and Probability. Wadsworth, Brooks and Cole, 1989

Halmos, Paul R. (1950). Measure theory. D. van Nostrand Co. See especially Sect. 51 Borel sets and Bairesets.

Halsey Royden, Real Analysis, Prentice Hall, 1988

Alexander S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995 (Graduate texts in Math., vol.156)
https://en.wikipedia.org/wiki/Irrational_numberhttps://en.wikipedia.org/wiki/Continued_fractionhttps://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Subsequencehttps://en.wikipedia.org/wiki/Divisorhttps://en.wikipedia.org/wiki/Analytic_sethttps://en.wikipedia.org/wiki/Descriptive_set_theoryhttps://en.wikipedia.org/wiki/Alexander_S._Kechrishttps://en.wikipedia.org/wiki/Parity_function#Infinite_parity_functionhttps://en.wikipedia.org/wiki/Halmoshttps://en.wikipedia.org/wiki/Borel_set#CITEREFHalmos1950https://en.wikipedia.org/wiki/Sigma-ringhttps://en.wikipedia.org/wiki/Saturated_sethttps://en.wikipedia.org/wiki/Second_countablehttps://en.wikipedia.org/wiki/Baire_sethttps://en.wikipedia.org/wiki/Cylindrical_%CF%83-algebrahttps://en.wikipedia.org/wiki/Polish_spacehttps://en.wikipedia.org/wiki/Descriptive_set_theoryhttps://en.wikipedia.org/wiki/Borel_hierarchyhttps://en.wikipedia.org/wiki/William_Arvesonhttps://en.wikipedia.org/wiki/Richard_Dudleyhttps://en.wikipedia.org/wiki/Paul_Halmoshttps://en.wikipedia.org/wiki/Halsey_Roydenhttps://en.wikipedia.org/wiki/Alexander_S._Kechris

10 CHAPTER 2. BOREL SET

2.7 Notes[1] Mackey, G.W. (1966), Ergodic Theory and Virtual Groups, Math. Annalen. (Springer-Verlag) 166 (3): 187207,

doi:10.1007/BF01361167, ISSN 0025-5831, (subscription required (help))

[2] Jochen Wengenroth (mathoverflow.net/users/21051), Is every sigma-algebra the Borel algebra of a topology?, http://mathoverflow.net/questions/87888 (version: 2012-02-09)

[3] Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 0-387-98412-7

[4] Lusin, Nicolas (1927), Sur les ensembles analytiques, Fundamenta Mathematicae (Institute of mathematics, Polishacademy of sciences) 10: 195.

[5] Tommy Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in: Probability and Lattices, in: CWI Tract, vol.110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150

2.8 External links Hazewinkel, Michiel, ed. (2001), Borel set, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

Formal definition of Borel Sets in the Mizar system, and the list of theorems that have been formally provedabout it.

Weisstein, Eric W., Borel Set, MathWorld.
https://en.wikipedia.org/wiki/George_Mackeyhttp://link.springer.com/article/10.1007%252FBF01361167https://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252FBF01361167https://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0025-5831http://mathoverflow.net/questions/87838/is-every-sigma-algebra-the-borel-algebra-of-a-topologyhttp://mathoverflow.net/questions/87838/is-every-sigma-algebra-the-borel-algebra-of-a-topologyhttps://en.wikipedia.org/wiki/Springer_Verlaghttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-387-98412-7http://www.encyclopediaofmath.org/index.php?title=p/b017120https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4http://mws.cs.ru.nl/mwiki/prob_1.html#K12https://en.wikipedia.org/wiki/Mizar_systemhttp://mmlquery.mizar.org/cgi-bin/mmlquery/emacs_search?input=(symbol+Borel_Sets+%257C+notation+%257C+constructor+%257C+occur+%257C+th)+ordered+by+number+of+refhttps://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://mathworld.wolfram.com/BorelSet.htmlhttps://en.wikipedia.org/wiki/MathWorld

Chapter 3

Cyclic order

In mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory,a cyclic order is not modeled as a binary relation, such as "a < b". One does not say that east is more clockwise thanwest. Instead, a cyclic order is defined as a ternary relation [a, b, c], meaning after a, one reaches b before c. Forexample, [June, October, February]. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive,

11
https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Circlehttps://en.wikipedia.org/wiki/Cyclic_order#endnote_cyclic_orderhttps://en.wikipedia.org/wiki/Order_theoryhttps://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Ternary_relationhttps://en.wikipedia.org/wiki/Cyclic_order#The_ternary_relationhttps://en.wikipedia.org/wiki/Cyclic_order#The_ternary_relation

12 CHAPTER 3. CYCLIC ORDER

and total. Dropping the total requirement results in a partial cyclic order.A set with a cyclic order is called a cyclically ordered set or simply a cycle.[nb] Some familiar cycles are discrete,having only a finite number of elements: there are seven days of the week, four cardinal directions, twelve notes inthe chromatic scale, and three plays in rock-paper-scissors. In a finite cycle, each element has a next element anda previous element. There are also continuously variable cycles with infinitely many elements, such as the orientedunit circle in the plane.Cyclic orders are closely related to the more familiar linear orders, which arrange objects in a line. Any linear ordercan be bent into a circle, and any cyclic order can be cut at a point, resulting in a line. These operations, along with therelated constructions of intervals and covering maps, mean that questions about cyclic orders can often be transformedinto questions about linear orders. Cycles have more symmetries than linear orders, and they often naturally occur asresidues of linear structures, as in the finite cyclic groups or the real projective line.

3.1 Finite cycles

V1 V2

V3

V4

V5

A 5-element cycle

A cyclic order on a set X with n elements is like an arrangement of X on a clock face, for an n-hour clock. Eachelement x in X has a next element and a previous element, and taking either successors or predecessors cycles
https://en.wikipedia.org/wiki/Cyclic_order#The_ternary_relationhttps://en.wikipedia.org/wiki/Cyclic_order#The_ternary_relationhttps://en.wikipedia.org/wiki/Partial_cyclic_orderhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Cyclic_order#endnote_cyclehttps://en.wikipedia.org/wiki/Finite_sethttps://en.wikipedia.org/wiki/Element_(mathematics)https://en.wikipedia.org/wiki/Days_of_the_weekhttps://en.wikipedia.org/wiki/Cardinal_directionhttps://en.wikipedia.org/wiki/Chromatic_scalehttps://en.wikipedia.org/wiki/Rock-paper-scissorshttps://en.wikipedia.org/wiki/Unit_circlehttps://en.wikipedia.org/wiki/Linear_orderhttps://en.wikipedia.org/wiki/Line_(geometry)https://en.wikipedia.org/wiki/Finite_cyclic_grouphttps://en.wikipedia.org/wiki/Real_projective_line

3.2. DEFINITIONS 13

exactly once through the elements as x(1), x(2), ..., x(n).There are a few equivalent ways to state this definition. A cyclic order on X is the same as a permutation that makes allof X into a single cycle. A cycle with n elements is also a Zn-torsor: a set with a free transitive action by a finite cyclicgroup.[1] Another formulation is to make X into the standard directed cycle graph on n vertices, by some matchingof elements to vertices.It can be instinctive to use cyclic orders for symmetric functions, for example as in

xy + yz + zx

where writing the final monomial as xz would distract from the pattern.A substantial use of cyclic orders is in the determination of the conjugacy classes of free groups. Two elements g andh of the free group F on a set Y are conjugate if and only if, when they are written as products of elements y and y1with y in Y, and then those products are put in cyclic order, the cyclic orders are equivalent under the rewriting rulesthat allow one to remove or add adjacent y and y1.A cyclic order on a set X can be determined by a linear order on X, but not in a unique way. Choosing a linear orderis equivalent to choosing a first element, so there are exactly n linear orders that induce a given cyclic order. Sincethere are n! possible linear orders, there are (n 1)! possible cyclic orders.

3.2 Definitions

An infinite set can also be ordered cyclically. Important examples of infinite cycles include the unit circle, S1, andthe rational numbers, Q. The basic idea is the same: we arrange elements of the set around a circle. However, inthe infinite case we cannot rely upon an immediate successor relation, because points may not have successors. Forexample, given a point on the unit circle, there is no next point. Nor can we rely upon a binary relation to determinewhich of two points comes first. Traveling clockwise on a circle, neither east or west comes first, but each followsthe other.Instead, we use a ternary relation denoting that elements a, b, c occur after each other (not necessarily immediately)as we go around the circle. For example, in clockwise order, [east, south, west]. By currying the arguments of theternary relation [a, b, c], one can think of a cyclic order as a one-parameter family of binary order relations, calledcuts, or as a two-parameter family of subsets of K, called intervals.

3.2.1 The ternary relation

The general definition is as follows: a cyclic order on a set X is a relation C X3, written [a, b, c], that satisfies thefollowing axioms:[nb]

1. Cyclicity: If [a, b, c] then [b, c, a]

2. Asymmetry: If [a, b, c] then not [c, b, a]

3. Transitivity: If [a, b, c] and [a, c, d] then [a, b, d]

4. Totality: If a, b, and c are distinct, then either [a, b, c] or [c, b, a]

The axioms are named by analogy with the asymmetry, transitivity, and totality axioms for a binary relation, whichtogether define a strict linear order. Edward Huntington (1916, 1924) considered other possible lists of axioms,including one list that was meant to emphasize the similarity between a cyclic order and a betweenness relation. Aternary relation that satisfies the first three axioms, but not necessarily the axiom of totality, is a partial cyclic order.

3.2.2 Rolling and cuts

Given a linear order < on a set X, the cyclic order on X induced by < is defined as follows:[2]

[a, b, c] if and only if a < b < c or b < c < a or c < a < b
https://en.wikipedia.org/wiki/Permutationhttps://en.wikipedia.org/wiki/Cycles_and_fixed_pointshttps://en.wikipedia.org/wiki/Torsorhttps://en.wikipedia.org/wiki/Group_actionhttps://en.wikipedia.org/wiki/Finite_cyclic_grouphttps://en.wikipedia.org/wiki/Finite_cyclic_grouphttps://en.wikipedia.org/wiki/Directed_cycle_graphhttps://en.wikipedia.org/wiki/Symmetric_functionhttps://en.wikipedia.org/wiki/Monomialhttps://en.wikipedia.org/wiki/Conjugacy_classhttps://en.wikipedia.org/wiki/Free_grouphttps://en.wikipedia.org/wiki/Rewritinghttps://en.wikipedia.org/wiki/Infinite_sethttps://en.wikipedia.org/wiki/Unit_circlehttps://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Curryinghttps://en.wikipedia.org/wiki/Cyclic_order#endnote_ternary_relationhttps://en.wikipedia.org/wiki/Asymmetric_relationhttps://en.wikipedia.org/wiki/Transitive_relationhttps://en.wikipedia.org/wiki/Total_relationhttps://en.wikipedia.org/wiki/Strict_linear_orderhttps://en.wikipedia.org/wiki/Edward_Vermilye_Huntingtonhttps://en.wikipedia.org/wiki/Cyclic_order#CITEREFHuntington1916https://en.wikipedia.org/wiki/Cyclic_order#CITEREFHuntington1924https://en.wikipedia.org/wiki/Betweenness_relationhttps://en.wikipedia.org/wiki/Partial_cyclic_order


Two linear orders induce the same cyclic order if they can be transformed into each other by a cyclic rearrangement,as in cutting a deck of cards.[3] One may define a cyclic order relation as a ternary relation that is induced by a strictlinear order as above.[4]

Cutting a single point out of a cyclic order leaves a linear order behind. More precisely, given a cyclically orderedset (K, [ ]), each element a K defines a natural linear order

3.4. FURTHER CONSTRUCTIONS 15

3.3.1 Functions on finite sets

A cyclic order on a finite set X can be determined by an injection into the unit circle, X S1. There are many possiblefunctions that induce the same cyclic orderin fact, infinitely many. In order to quantify this redundancy, it takesa more complex combinatorial object than a simple number. Examining the configuration space of all such mapsleads to the definition of an (n 1)-dimensional polytope known as a cyclohedron. Cyclohedra were first applied tothe study of knot invariants;[11] they have more recently been applied to the experimental detection of periodicallyexpressed genes in the study of biological clocks.[12]

The category of homomorphisms of the standard finite cycles is called the cyclic category; it may be used to constructAlain Connes' cyclic homology.One may define a degree of a function between cycles, analogous to the degree of a continuous mapping. For example,the natural map from the circle of fifths to the chromatic circle is a map of degree 7. One may also define a rotationnumber.

3.3.2 Completion

A cut with both a least element and a greatest element is called a jump. For example, every cut of a finite cycleZn is a jump. A cycle with no jumps is called dense.[13][14]

A cut with neither a least element nor a greatest element is called a gap. For example, the rational numbers Qhave a gap at every irrational number. They also have a gap at infinity, i.e. the usual ordering. A cycle with nogaps is called complete.[15][14]

A cut with exactly one endpoint is called a principal or Dedekind cut. For example, every cut of the circle S1 isa principal cut. A cycle where every cut is principal, being both dense and complete, is called continuous.[16][14]

The set of all cuts is cyclically ordered by the following relation: [


[

3.5. TOPOLOGY 17

c = a b and z < x

a = b = c and [x, y, z]

The lexicographic product K L globally looks like K and locally looks like L; it can be thought of as K copies of L.This construction is sometimes used to characterize cyclically ordered groups.[21]

One can also glue together different linearly ordered sets to form a circularly ordered set. For example, given twolinearly ordered sets L1 and L2, one may form a circle by joining them together at positive and negative infinity. Acircular order on the disjoint union L1 L2 {, } is defined by < L1 < < L2 < , where the inducedordering on L1 is the opposite of its original ordering. For example, the set of all longitudes is circularly ordered byjoining together all points west and all points east, along with the prime meridian and the 180th meridian. Kuhlmann,Marshall & Osiak (2011) use this construction while characterizing the spaces of orderings and real places of doubleformal Laurent series over a real closed field.[22]

3.5 Topology

The open intervals form a base for a natural topology, the cyclic order topology. The open sets in this topology areexactly those sets which are open in every compatible linear order.[23] To illustrate the difference, in the set [0, 1),the subset [0, 1/2) is a neighborhood of 0 in the linear order but not in the cyclic order.Interesting examples of cyclically ordered spaces include the conformal boundary of a simply connected Lorentzsurface[24] and the leaf space of a lifted essential lamination of certain 3-manifolds.[25] Discrete dynamical systemson cyclically ordered spaces have also been studied.[26]

The interval topology forgets the original orientation of the cyclic order. This orientation can be restored by enrich-ing the intervals with their induced linear orders; then one has a set covered with an atlas of linear orders that arecompatible where they overlap. In other words, a cyclically ordered set can be thought of as a locally linearly orderedspace: an object like a manifold, but with order relations instead of coordinate charts. This viewpoint makes it easierto be precise about such concepts as covering maps. The generalization to a locally partially ordered space is studiedin Roll (1993); see also Directed topology.

3.6 Related structures
https://en.wikipedia.org/wiki/Longitudehttps://en.wikipedia.org/wiki/Prime_meridianhttps://en.wikipedia.org/wiki/180th_meridianhttps://en.wikipedia.org/wiki/Cyclic_order#CITEREFKuhlmannMarshallOsiak2011https://en.wikipedia.org/wiki/Cyclic_order#CITEREFKuhlmannMarshallOsiak2011https://en.wikipedia.org/wiki/Real_placehttps://en.wikipedia.org/wiki/Formal_Laurent_serieshttps://en.wikipedia.org/wiki/Real_closed_fieldhttps://en.wikipedia.org/wiki/Base_(topology)https://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Order_topologyhttps://en.wikipedia.org/wiki/Open_sethttps://en.wikipedia.org/wiki/Simply_connectedhttps://en.wikipedia.org/wiki/Lorentz_surfacehttps://en.wikipedia.org/wiki/Lorentz_surfacehttps://en.wikipedia.org/wiki/Leaf_spacehttps://en.wikipedia.org/wiki/Essential_laminationhttps://en.wikipedia.org/wiki/Discrete_dynamical_systemhttps://en.wikipedia.org/wiki/Manifoldhttps://en.wikipedia.org/wiki/Cyclic_order#CITEREFRoll1993https://en.wikipedia.org/wiki/Directed_topology


3.6.1 Groups

Main article: Cyclically ordered group

A cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplicationboth preserve the cyclic order. Cyclically ordered groups were first studied in depth by Ladislav Rieger in 1947.[27]They are a generalization of cyclic groups: the infinite cyclic group Z and the finite cyclic groups Z/n. Since a linearorder induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rationalnumbers Q, the real numbers R, and so on. Some of the most important cyclically ordered groups fall into neitherprevious category: the circle group T and its subgroups, such as the subgroup of rational points.Every cyclically ordered group can be expressed as a quotient L / Z, where L is a linearly ordered group and Z is acyclic cofinal subgroup of L. Every cyclically ordered group can also be expressed as a subgroup of a product T L,where L is a linearly ordered group. If a cyclically ordered group is Archimedean or compact, it can be embedded inT itself.[28]

3.6.2 Modified axioms

A partial cyclic order is a ternary relation that generalizes a (total) cyclic order in the same way that a partial ordergeneralizes a total order. It is cyclic, asymmetric, and transitive, but it need not be total. An order variety is a partialcyclic order that satisfies an additional spreading axiom. Replacing the asymmetry axiom with a complementaryversion results in the definition of a co-cyclic order. Appropriately total co-cyclic orders are related to cyclic ordersin the same way that is related to

3.9. NOTES ON USAGE 19

3.9 Notes on usage

^cyclic order The relation may be called a cyclic order (Huntington 1916, p. 630), a circular order (Huntington1916, p. 630), a cyclic ordering (Kok 1973, p. 6), or a circular ordering (Mosher 1996, p. 109). Some authors callsuch an ordering a total cyclic order (Isli & Cohn 1998, p. 643), a complete cyclic order (Novk 1982, p. 462), alinear cyclic order (Novk 1984, p. 323), or an l-cyclic order or -cyclic order (ernk 2001, p. 32), to distinguishfrom the broader class of partial cyclic orders, which they call simply cyclic orders. Finally, some authors may takecyclic order to mean an unoriented quaternary separation relation (Bowditch 1998, p. 155).^cycle A set with a cyclic order may be called a cycle (Novk 1982, p. 462) or a circle (Giraudet & Holland 2002,p. 1). The above variations also appear in adjective form: cyclically ordered set (cyklicky uspodan mnoiny, ech1936, p. 23), circularly ordered set, total cyclically ordered set, complete cyclically ordered set, linearly cyclicallyordered set, l-cyclically ordered set, -cyclically ordered set. All authors agree that a cycle is totally ordered.^ternary relation There are a few different symbols in use for a cyclic relation. Huntington (1916, p. 630) usesconcatenation: ABC. ech (1936, p. 23) and (Novk 1982, p. 462) use ordered triples and the set membershipsymbol: (a, b, c) C. Megiddo (1976, p. 274) uses concatenation and set membership: abc C, understanding abcas a cyclically ordered triple. The literature on groups, such as wierczkowski (1959a, p. 162) and ernk & Jakubk(1987, p. 157), tend to use square brackets: [a, b, c]. Giraudet & Holland (2002, p. 1) use round parentheses: (a, b,c), reserving square brackets for a betweenness relation. Campero-Arena & Truss (2009, p. 1) use a function-stylenotation: R(a, b, c). Rieger (1947), cited after Pecinov 2008, p. 82) uses a less-than symbol as a delimiter: < x,y, z


[14] Novk & Novotn 1987, p. 409410.

[15] Novk 1984, pp. 325, 331.

[16] Novk 1984, p. 333.

[17] Novk 1984, p. 330.

[18] Roll 1993, p. 469; Freudenthal & Bauer 1974, p. 10

[19] Freudenthal 1973, p. 475; Freudenthal & Bauer 1974, p. 10

[20] wierczkowski 1959a, p. 161.

[21] wierczkowski 1959a.

[22] Kuhlmann, Marshall & Osiak 2011, p. 8.

[23] Viro et al. 2008, p. 44.

[24] Weinstein 1996, pp. 8081.

[25] Calegari & Dunfield 2003, pp. 1213.

[26] Bass et al. 1996, p. 19.

[27] Pecinov-Kozkov 2005, p. 194.

[28] wierczkowski 1959a, pp. 161162.

[29] Knuth 1992, p. 4.

[30] Huntington 1935.

[31] Macpherson 2011.

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Calegari, Danny; Dunfield, Nathan M. (April 2003), Laminations and groups of homeomorphisms of thecircle, Inventiones Mathematicae 152 (1): 149204, arXiv:math/0203192, doi:10.1007/s00222-002-0271-6

Campero-Arena, G.; Truss, John K. (April 2009), 1-transitive cyclic orderings (PDF), Journal of Combina-torial Theory, Series A 116 (3): 581594, doi:10.1016/j.jcta.2008.08.006, retrieved 25 April 2011

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Ring of sets 1_8