Space and Time - univie.ac.at

Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)

© Wolfgang Kainz 1

Mathematical Principles

of GISWolfgang Kainz

Department of Geography and Regional Research

University of Vienna, Austria

Contents

• Spatial information• History of GIS• GI-Science• Mathematical methods

© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 2

© Wolfgang Kainz 3

Space and Time

Department of Geography and Regional Research, University of Vienna

Space and Time

Creation myths start with the creation of space and time (often out of chaos).Then comes the rest…

Can we imagine something without a connection to space and time?


We are spatiotemporal beings

• Limited to three spatial dimensions– We cannot escape from within a closed cube

(we cannot “see” higher dimensions than 3D) like

– Flatlanders cannot escape from a closed square


Time…(Augustine)

• “…For what is time? Who can easily and briefly explain it? Who can even comprehend it in thought or put the answer into words? Yet is it not true that in conversation we refer to nothing more familiarly or knowingly than time? And surely we understand it when we speak of it; we understand it also when we hear another speak of it. …

• What, then, is time? If no one asks me, I know what it is. If I wish to explain it to him who asks me, I do not know.”



© Wolfgang Kainz 2

Importance of Space and Time

• Almost everything that happens, happens at a certain location in space and time

• The level of (geographic) detail (or scale) matters– Mapping a local event versus the global climatic change

• Time scales– 500-year flood versus property transactions


Why is spatial special?

• We are dealing with multiple dimensions (x,y,z)• We are dealing with different levels of spatial

resolution• Representation of spatial data is more “complicated”

than of non-spatial data• We often need to transform and project data• Spatial analysis requires special methods



GI Science

and Systems data

information

knowledge

wisdom

acquisition

analysis

reasoning

contemplation

Disciplines Using Spatial Information


Type of discipline Sample disciplines Development of spatial concepts

Geography, cartography, cognitive science, linguistics, psychology, philosophy

Means for capturing and processing of spatial data

Remote sensing, surveying engineering, cartography, photogrammetry

Formal and theoretical foundation

Computer science, knowledge based systems, mathematics, statistics

Applications Archaeology, architecture, forestry, geo-sciences, regional and urban planning, surveying

Support Law, economy

First, there were systems…

• Development of geographic information systems– A special type of information system dealing with

geographic information (spatial information)– Application of information technology

© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 11 © Wolfgang Kainz 12

History of GIS



© Wolfgang Kainz 3

Description of the Earth

© Wolfgang Kainz 13

Land administration Geodesy & Geometry

Sumerians, Babylonians,Egyptians

Greeks

Euclid, Eratosthenes,Ptolemy


Ga-Sur Tablet (3800 BC)

© Wolfgang Kainz 14Department of Geography and Regional Research, University of Vienna

City Map of Nippur (1500 BC)

© Wolfgang Kainz 15Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz 16

Pocket-GIS



The Early Years


The Early Years (1965 – 1985)

• Canada GIS• Insufficient hardware• Experimental software• “Discovery” of topology• Relational databases• Problems with acceptance and understanding

(“map data models”)• No integration of GIS and RS



© Wolfgang Kainz 4

GIS Software

Data StructuresAlgorithms

Data Input Database

Output &Visualization

Analysis


Vector: Arc/NodeStructure

Raster: Quadtree

Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz 20

Consolidation


Consolidation (1985 – 1992)

• Introduction of PCs and minicomputers• Functional software• GIS in central and local government

organizations• GIS in private industry• GIS in education• Textbooks and journals• Functional GIS and RS software (separate)

© Wolfgang Kainz 21Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz 22

1986 1987Department of Geography and Regional

Research, University of Vienna

GIS Software

Data StructuresAlgorithms

Data Input Database


Analysis


SpatialModeling

GIS and Decision Support Systems

Management and Infrastructure

Theoretical Foundation

Data Input Database


Analysis


• Topology• Integration of vector- and

raster data• Separation of attribute and

geometry data



© Wolfgang Kainz 5


Operationalization


Operationalization (1992 – 2000)

• GIS in business and science• Desire for a theoretical foundation

– Geomatics/geoinformatics– Geographic Information Science

• Spatial Modeling– Theory of spatial relations– Ontologies

• Integration of GIS and RS


Then, came the science…

• The science behind the systems– Geoinformatics– Geomatics– Spatial information science– Spatial information theory– Geoinformation engineering– Geographic Information Science


Space and Time in GIS


formal & theoretical aspects conceptual & empirical aspects

mathematics physics philosophy geography

Euclidean space

metric space

topological space

space-time absolute vs. relative

atomic vs. plenum

object vs. field

cognitive dim.

Democritus, Parmenides, Newton, LeibnizKant, Wittgenstein, LakeoffEinstein, Hawkins


SpatialModeling


Management and Infrastructure


Data Input Database


Analysis


Geographic Information Science

SpatialModeling


Management und Infrastructure


Data Input Database


Analysis


• Theory of spatialrelations

• Integration of vectorand raster data

• Integration of attributeand geometry data(geodatabase)



© Wolfgang Kainz 6


Spatial Modeling


Models

• Spatial modeling• Implementation of models• Application of models


Modeling

• 2.12 “A picture is a model of reality.”• 2.14 “What constitutes a picture is that its

elements are related to one another in a determinate way.”

• 2.15 “The fact that the elements of a picture are related to one another in a determinate way represents that things are related to one another in the same way.”


(Wittgenstein: Tractatus Logico-Philosophicus)

Spatial Modeling & Data Processing


• Real world phenomena have a spatiotemporal extent and possess thematic characteristics (attributes).

• A (spatial) feature is a representation of a real world phenomenon.

• Spatial data are computer representations of spatial features.

• Spatial data handling extracts (spatial) informationfrom spatial data.

Real Worldphenomena

Spatial modelfeatures

GIS databasespatial data

design implementation

Data handling

Spatial Information


Real World

Miniworld

Conceptualschema

Logicalschema

Physicalschema

Conceptual model

Logical model

Implementation

software independent

software specific

Spatial modeling

Spat

ial

mod

elin

gDa

taba

se d

esig

nan

d im

plem

enta

tion

Ontology

Studies being or existence and their basic categories and relationships, to determine what entities and what types of entities exist. Ontology thus has strong implications for conceptions of reality.



© Wolfgang Kainz 7

GIScience is

• Ontology driven– What constitutes the world?

• Entities (objects, categories, concepts)• Characteristics (attributes)

– How are things related?• Relations (relationships)


GIScience uses

• Logic (language of mathematics)• Mathematics (structures)

To make statements about the world and acquire knowledge about the world



Mathematics

Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz Department of Geography and Regional

Research, University of Vienna 40

Structures

Algebraic Order Topological

Logic

Set Theory

RelationsFunctions

Algebra OrderedSets Topology


Logic


Logic

• Propositional Logic– propositions, logical operators (and, or, not, …)

• Predicate Logic– Predicates (properties or relations), quantifiers

• Logical Inference– Rules of inference



© Wolfgang Kainz 8


Assertion and Proposition

• An assertion is a statement.– “Are you okay?”– “Give me that book.”

• A proposition is an assertion that is either true or false, but not both.– “It is raining.”– “I pass the exam.”


Propositional Variable and Propositional Form

• A propositional variable is a proposition with an unspecified truth value denoted as P, Q, R, …, etc.

• An assertion with at least one propo-sitional variable is called a propositional form, e.g., Pand “I pass the exam.” When propositions are substituted for the variables a proposition results.



Logical Operators

• Propositions and propositional variables can be combined with logical operators (or logical connectives) to form new assertions. Variables are called operands.

• Operators: not (), and (), or (), exclusive or (), implication (), equivalence ()

• "not P and Q " or "P Q "• "I study hard and I pass the exam."


Truth Tables

• Truth tables show the truth values for all possible combinations of true and false for the operands.

• We use 0 for “false” and 1 for “true”.



Negation

01

10

PP


Conjunction(“logical and”)

111

001

010

000

QPQP



© Wolfgang Kainz 9


Disjunction(“logical or” or “inclusive or”)

111

101

110

000

QPQP


Exclusive or

011

101

110

000

QPQP


Implication


P is premise, hypothesis, or antecedent, Q is conclusion or consequence.

– "If P then Q."– "P only if Q."– "Q if P."– "P is a sufficient condition for Q."– "Q is a necessary condition for P."

111

001

110

100

QPQP

Equivalence


• “P is equivalent to Q ”• “P is a necessary and

sufficient condition for Q ”

• “P if and only if Q ” or “P iff Q ”

111

001

010

100

QPQP


Types of Propositional Forms

• A tautology is a propositional form whose truth value is true for all possible values of its propositional variables.

• A contradiction (or absurdity) is a propositional form which is always false.

• A contingency is a propositional form which is neither a tautology nor a contradiction.


Examples


(P Q) P is a tautology.

1111

1001

1010

1000

)( PQPQPQP



Examples


P P is a contradiction.

001

010

PPPP

Examples


(P Q) Q is a contingency.

11011

01101

10010

01100

)( QQPQPQQP


Predicates

Predicates express a property of an object or a relationship between objects. Objects are often represented by variables.

• “x lives in y ” written as L(x,y). Here, x and y are variables, L or “lives in” is a predicate. L is said to have two arguments, x and y, or to be a 2-place predicate.

• Also, “x is equal to y” or “x = y ”, and “x is greater than y ” or “x > y ” are 2-place predicates.


Predicates

• Values for variables must be taken from a set, the universe of discourse (or universe).

• To change a predicate into a proposition, each individual variable must be bound by either assigning a value to it, or by quantification of the variable.



Universal Quantifier

Universal quantifier . It is read “for all”, “for every”, “for any”, “for arbitrary”, or “for each”.

• “For all x, P(x) “ or “xP(x) ” is interpreted as “For all values of x, the assertion P(x) is true.”


If an assertion P(x) is true for every possible value x, then xP(x) is true; otherwise xP(x) is false.


Existential Quantifier

Existential quantifier . It is read as “there exists”, “for some”, or “for at least one”. A variation ! means “there exists a unique x such that …” or "there is one and only one x such that …".

“For some x, P(x) ” or “xP(x) ” is interpreted as “There exists a value of x for which the assertion P(x) is true.”


If an assertion P(x) is true for at least one value x, then xP(x) is true; otherwise xP(x) is false.




Logical Inference

• A theorem is a mathematical assertion which can be shown to be true.

• A proof is an argument which establishes the truth of the theorem.


Logical Inference


Rules of inference specify conclusions which can be drawn from assertions known or assumed to be true.

QP

PP

n

2

1

The assertions Pi are called hypotheses or premises,the assertion below the line is called conclusion. Thesymbol is read “therefore” or “it follows that” or“hence.”


Logical Inference

• An argument is said to be valid or correct if, whenever all the premises are true, the conclusion is true.

• An argument is correct when (P1 P2 … Pn) Q is a tautology.

Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz Department of Geography and Regional

Research, University of Vienna 64

Reasoning: Rules of inference

In binary logic reasoning is based on – Deduction (modus ponens)

• Premise 1: If x is A then y is B• Premise 2: x is A• Conclusion: y is B

– Induction (moduls tollens)• Premise 1: If x is A then y is B• Premise 2: y is not B• Conclusion: x is not A


Reasoning: Rules of inference

Example– Deduction (modus ponens)

• Premise 1: If it rains then I get wet• Premise 2: It rains• Conclusion: I get wet

– Induction (moduls tollens)• Premise 1: If it rains then I get wet• Premise 2: I do not get wet• Conclusion: It does not rain


Generalized modus ponens

1 1

2 2

1

1

If is then is If is then is

:

If is then is : is : is

n n

x A y Bx A y B

p q

x A y Bp x Aq y B




AlgebraicStructures


Algebraic Structures

• Sets, elements, operators– Arithmetic– Map algebra– Relational algebra

• Structure preserving mappings– Homomorphism– Isomorphism



Components of an Algebra

• A set, the carrier of the algebra,• Operations defined on the carrier, and• Distinguished elements of the carrier, the

constants of the algebra.

• Algebras are presented as tupels <carrier, operations, constants>– Example: <R,+,,0,1>


Constants of Algebras

• Let be a binary operation on S.• An element 1S is an identity (or unit) for the

operation if 1 x = x 1 = x for every x in S.• An element 0S is a zero for the operation if

0 x = x 0 = 0 for every x in S.These constants are also called identity element and zero

element, respectively.

Examples: 1 is an identity element and 0 is a zero element for the multiplication of numbers. The number 0 is a identity element for the addition of numbers.



Constants of Algebras

• Let be a binary operation on S and 1 an identity for the operation.

• If x y = 1 and y x = 1 for every y in S, then x is called a (two-sided) inverse of y with respect to the operation .

Example: In the real numbers 1/x is the inverse with respect to multiplication.


Group


A group is an algebra with the signature <S,, ̄,1>, with ̄ the inverse with respect to , and the following axioms:

1

11

)()(

aaaaa

cbacba

If the operation is also commutative, we call the group a commutative group.




Example

• <I,+,-,0> is a group where I are the integers, “+” is the addition, “-” the inverse (negative) integer, and 0 the identity for the addition.

• <R-{0},,-1,1> is a group where R are the real numbers, “” is the multiplication, “-1” the inverse, and 1 the identity for the multiplication.


Field

A field is an algebra with the signature<F,+,, ¯,-1,0,1> and the following axioms:

1. <F,+, ¯,0> is a commutative group2. a (b c) = (a b) c3. a (b + c) = a b + a c4. (a + b) c = a c + b c5. <F-{0},,-1,1> is a commutative group



Example

The real numbers <R,+,,-, -1,0,1> are a field with the addition and multiplication as binary operations, and the inverse unary operations for the addition and multiplication. The numbers 0 and 1 function as unit elements for + and , respectively.


Boolean Algebra

A Boolean algebra is an algebra with signature <S,+,, ¯,0,1>, where + and are binary operations, and ¯ is a unary operation (complementation), with the following axioms:



Boolean Algebra

complement the ofproperty 0

complement the ofproperty 1

foridentity an is 11

foridentity an is 00

law vedistributi)()()(

law vedistributi)(

law eassociativ)()(

law eassociativ)()(

law ecommutativ

law ecommutativ

aaaa

aaaa

cabacbacabacba

cbacbacbacba

abbaabba


Example

<P(A),,, ¯,{},A> with ¯ as the complement relative to A, is a Boolean algebra.




Vector Space


Let <V,+, ¯,0> be a commutative group, and<F,+,, ¯ ,-1,0,1> a field. V is called a vector spaceover F if for all a, b V and , F

aaaa

aaababa

1

)()(

)(

)(


Example

The set of all vectors V with + as the vector addition is a vector space over the real numbers R where is the multiplication of a vector with a scalar.The same is true for the set of all matrices M with the matrix addition and the multiplication of a matrix with a scalar.


Isomorphism


Two algebras <S,,,k> and <S’,’,’,k’> are isomorphic if there exists a bijection f such that

kkfafaf

bfafbafSSf

)()4(

))(())(()3(

)()()()2(

:)1(


Isomorphism

• Two isomorphic algebras are essentially the same structure with different names.

• If we do not require f to be a bijection then we talk about a homomorphism. In general, homomorphisms generate a “smaller” image of an algebra of the same class.



TopologicalStructures


Topological Structures

• “Good behavior” of neighborhoods of points and invariants– Certain (spatial) relationships of neighborhood,

connectivity– Simple structured spaces (simplexes, cells)– Spatial relations derived from topological

invariants• Structure preserving mappings

– Topological mapping (or homeomorphism)




Topology

• Metric space• Neighborhood• Topological space• Homeomorphism• Simplices, cells and their complexes


What is Topology?

• Topology is the study of certain invariants of structured spaces.

• Structuring of spaces through:– a generalized notion of distance (metric space),– abstract notion of neighborhood, or– pasting together of certain well-understood

elementary objects (simplices or cells) to complexes (simplicial or cell complex).


Spaces used in GIS

• In GIS we normally deal with objects in the Euclidean space in one, two, or three dimensions (R1, R2, or R3).

• Objects in this space are represented by nodes, arcs, polygons, and volumes.


Properties of Space for Spatial Data

• Three-dimensional Euclidean space• Vector space• Metric space (Euclidean metric)• Topological space (topology induced by

Euclidean metric)• The topological space is structured by simple

sub-spaces (simplexes, cells, and their complexes)


Metric Space

),(),(),(

),(),(

ifonly and if0),( and 0),(

zydyxdzxdxydyxd

yxyxdyxd


Let M be a set and d : M x M R a function, the metric (or distance function) on M.(M ; d) is called a metric space if the following conditions are valid for all x, y, z from M :


Metric Space:Examples for Distance Functions


3 21

1 33

1

( , ) ( )

( , ) max | |

( , ) | |

( , ) 0, if , and ( , ) 1, otherwise

i ii

i i i

i ii

d x y x y

d x y x y

d x y x y

d x y x y d x y

Euclideanmetric

City blockmetric



Open Sets


N

M

xr

K

N is an open neighborhood ofx M, if there exists an open disk K(x,r) with r > 0 andK(x,r) N.

N is an open set, if N is an open neighborhood of each of its points.

Metric Space

• An open disk of radius r around x of M is defined asK (x,r) = {y M | d (x,y) < r}

• A set N is called an (open) neighborhood of a point xof M, if there exists an open disk K (x, r) around xsuch that K (x, r) N.

• A set N M is called open set, if N is an open neighborhood of each of its points.



x

x

U

N

x

N

V

(N1) The point x lies in each of its neighborhoods.

(N4) Every neighborhood N of x contains a neighborhood V of x such that N is a neighborhood of every point of V.

(N3) Every superset U of a neighborhood N of x is a neighborhood of x. X is a neighborhood of x.

(N2) The intersection of two neighbor-hoods of x is itself a neighborhood.

x1N

2N

Topological Space

• Let M be a set. A topology on M is a collection O of subsets of M with the following properties:

(O1) {} O, M O(O2) A, B O A B O(O3) Ai O for all i I Ui I Ai O

• The elements of O are called open sets• (O4) N is an (open) neighborhood of x M, if

x N O.• (M,O) is called a topological space.


Topological spacebased on concept ofneighborhood(N1 to N4):

Topological spaceBased on concept ofopen set(O1 to O3):

Definitionof open set

Definition ofneighborhood (O4)

Propertiesof open sets:O1 to O4

Propertiesof neighborhoods:N1 to N4

More theorems

Axioms

Theorems


( ,{ ( )| })X N x x X

Topological spacebased on concept ofneighborhood(N1 to N4):

Definitionof open set

Propertiesof open sets:O1 to O4

( , )X O

Topological spaceBased on concept ofopen set(O1 to O3):

Definition ofneighborhood (O4)

Propertiesof neighborhoods:N1 to N4

Homeomorphism

• A bijective, continuous mapping with continuous inverse between two topological spaces is called a homeomorphism.





Simplexes, Cells, Complexes

Simplexes, cells and their complexes are simple kinds of spaces that serve as topological equivalents of more complicated subsets of Euclidean space.


Simplexes


0-simplex (point)

0v1-simplex (closed line segment)

0v 1v

2-simplex (triangle)

2v

0v 1v

3-simplex (solid tetrahedron)

3v

0v 1v

2v


Simplex

• A p-dimensional simplex Sp is a “solid” polyhedron in Rn which has internal points, is convex, and has a minimal number of vertices.– For the dimensions 1, 2, and 3, we have straight

line segments, solid triangles, and solid tetrahedrons.

• A q-dimensional face of a simplex Sp is a q-dimensional subset (simplex) of the p-dimensional simplex.


Simplexes and Simplicial Complex


0-simplex

1-simplex

2-simplex

3-simplex

Simplicial complex


Simplicial Complex

A simplicial complex S is a set of simplexes in Rn

that fulfill the following conditions:– If the simplex Sp is an element of S, then each

face of Sp belongs to S.– For any two simplexes in S the intersection is

either empty or a common face.


Cell

• A p-dimensional cell (or p-cell ) is a set which is homeomorphic to the p-dimensional unit ball.Every (open) p -simplex is a p -cell.

• Cell complexes (or CW spaces) are built starting with 0-cells (nodes) and subsequently gluing 1-cells (arcs), 2-cells (polygons), etc.





0-dimensional unit cell




0-cell

1-cell

2-cell

3-cell

Cells and Cell Complex


0-cell

1-cell

2-cell

3-cell

Cell complex


2 2

Simplicial Complex Simplicial Complex


Cell decompositionof a 2-dimensional space

1-dimensional skeleton 0-dimensional skeleton

Cell Decomposition



Start with 0-cells

Gluing of 1-cells Gluing of 2-cells

0X

1X 2X

Generation of a cellcomplex


Topological Mapping


h1M2M

A

B

C

f

e

d

c

b

a

43

2

1

2

A

B

C

f

e

d

c

b

a

43

2

1

2




Topological Consistency Constraints


• Every 1-cell is bounded by two 0-cells.

• For every 1-cell there are two 2-cells (left and right polygon).

• Every 2-cell is bounded by a closed cycle of 0- and 1-cells.

• Every 0-cell is surrounded by a closed cycle of 1- and 2-cells.

• 1-cells intersect only in 0-cells.


A

B

C

f

e

d

c

b

a

43

2

1

2

Consistency Constraints: Example



A

B

C

f

e

d

c

b

a

43

2

1

2

Closed Boundary Criterion

A

B

C

e

d

c

b

3

2

1

A

B

C

e

d

c

b

3

2

1


A

B

C

f

e

d

c

b

a

43

2

1

2

Node Criterion (“Umbrella”)

A

B

C

f

e

d

c

b

a

43

2

1

2


Interior, closure, boundary, and exterior

• The interior of a set A (written as A°) is the union of all open sets contained in A.

• A is closed if its complement is open. The smallest closed set that contains A is the closure of A (written as ).

• The boundary of A (written as A) is the difference of the closure and the interior.

• The exterior of A (written as A¯) is the complement of the closure.


A

Topolocial Invariants


A

interior

A

boundary

A

closure

A

exterior



Topological Relationships


Relationships between two regions can be determined based on the intersection of their boundaries and interiors (4-intersection).

A B

BABABABABAI

),(4

Spatial Relationships Between Simple Regions


disjoint

meet

equal

inside

covered by

contains

covers

overlap


9-Intersection

––––

–

–

9 ),(

BABABABABABABABABA

BAI


Order Structures


Order Structures• Partially ordered set (poset) and lattice• Comparison of elements of a set

– Relationships of containment and inclusion• “… is contained in …”• “ … contains …”• “ … subset of …”• “ … less than or equal …”

– Representation of order diagrams as directed acyclic graph (DAG)

– Consideration of special elements (lower and upper bounds, greatest lower bound (g.l.b.), least upper bound (l.u.b.))

• Structure preserving mapping– Monotone functions


Partially ordered set

A partially ordered set (or poset) is a set P with a relation defined for its elements x, y, and z and:(1) x x (reflexive)(2) x y and y x implies x = y (antisymetric)(3) x y and y z imply x z (transitive)

Example: numbers with ( ), sets with ()




Order Relationships


A

B

CD

E

Order Relationships: Order Diagram


BCD

E

AA

B C

D E

Order Relationships: Order Diagram

• An order diagram is a DAG (directed acyclic graph).

• It can be represented by an adjacency matrix or list.

• Graph algorithms can be used.


Order Structures: Sample Questions

• What regions are contained in a set of given regions?

• What is the largest region contained in a set of given ones?


Upper and lower bounds

• Let P be a poset and S P. An elementx P is an upper bound of S if s x for alls S. A lower bound is defined by duality.

• The set of all upper or lower bounds are denoted as S* (S upper ) and S* (S lower ).


Greatest lower bound, least upper bound

• If S* has a largest element, it is called greatest lower bound (g.l.b.), meet, or infimum. For two elements x and y we write inf{x, y} or x y (“x meet y ”).

• If S* has a least element, it is called least upper bound (l.u.b.), join, or supremum. For two elements x and y we write sup{x, y} orx y (“x join y ”).




Example: lower bounds


A

B C

D E

Lower bounds of B are B, Dand E.

BB




A

B C

D E

Lower bounds of C are C, Dand E.

CC




A

B C

D ELower bounds of {B,C} are Dand E.

CBB C


Lattice

• A lattice is a poset in which a g.l.b. or l.u.b. can always be found for any two elements.

• If a g.l.b. or l.u.b. exists for every subset of the poset, we call it a complete lattice. Every finite lattice is complete.


Order Relationships: Normal Completion


poset

A

B C

D E

latticeA

B C

D E

X

{ }

Order Relationships: Geometric Interpretation


A

B C

D E

X

{ }

BCD

E

A

X



Monotone functions

• M and N are two posets. A functionf : MN is called a monotone function (or order preserving), if x y in M implies that f (x ) f (y ) in N.

• The function is an order-embedding when f is injective.

• If f is bijective and monotone with a monotone inverse, it is called an order isomorphism.


Problem Solving


Spatial problemTranslation into mathematical

problem

mathematical solution

Spatial interpretation

of solution

Solution of spatial problem


Find the largest region contained in

given regions

Generate the posetof regions

Compute the normal completion

and g.l.b.

Spatial interpretation of

g.l.b.Solution of spatial

problem

BCD

E

A A

B C

D E

A

B C

D E

X

{ }

BCD

E

A

X

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Space and Time - univie.ac.at