Mathematical relations qrstuzFrom Wikipedia, the free encyclopedia

Contents

1 Process calculus 11.1 Essential features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Mathematics of processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Parallel composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Sequential composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.4 Reduction semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.5 Hiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.6 Recursion and replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.7 Null process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Discrete and continuous process algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Current research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Software implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7 Relationship to other models of concurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Quasitransitive relation 62.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Quotient by an equivalence relation 83.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Rational consequence relation 10

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4.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Rational consequence relations via atom preferences . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.4.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 The representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.5.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Reduct 135.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Reflexive closure 146.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7 Reflexive relation 157.1 Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 Number of reflexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.4 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8 Relation algebra 198.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 Expressing properties of binary relations in RA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3 Expressive power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8.3.1 Q-Relation Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.5 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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8.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9 Relation construction 249.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10 Representation (mathematics) 2510.1 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

10.2.1 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2.2 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2.3 Polysemy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

10.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11 Separoid 2811.1 The axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.3 The basic lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

12 Series-parallel partial order 3012.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.2 Forbidden suborder characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.3 Order dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.4 Connections to graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.5 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

13 Surjective function 3413.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

13.3.1 Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3.4 Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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14 Symmetric closure 4014.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

15 Symmetric relation 4115.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

15.1.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4115.1.2 Outside mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

15.2 Relationship to asymmetric and antisymmetric relations . . . . . . . . . . . . . . . . . . . . . . . 4215.3 Additional aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

16 Ternary equivalence relation 4416.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

17 Ternary relation 4517.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

17.1.1 Binary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4517.1.2 Cyclic orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4517.1.3 Betweenness relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4517.1.4 Congruence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4517.1.5 Typing relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

17.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

18 Tolerance relation 4718.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

19 Total order 4819.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4819.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4919.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

19.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4919.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4919.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5019.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5019.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5019.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5019.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

19.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 5119.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5119.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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19.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5119.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

20 Total relation 5320.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5320.2 Properties and related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5320.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5320.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

21 Transitive closure 5521.1 Transitive relations and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.2 Existence and description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5621.4 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5621.5 In logic and computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5621.6 In database query languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5721.7 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5721.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5721.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5821.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

22 Transitive relation 5922.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

22.3.1 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6022.3.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6022.3.3 Properties that require transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

22.4 Counting transitive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6022.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6022.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

22.6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6122.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

22.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

23 Trichotomy (mathematics) 6223.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6223.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

24 Unimodality 6424.1 Unimodal probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

24.1.1 Other definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6624.1.2 Uses and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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24.1.3 Gauss’ inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6624.1.4 Vysochanskiï–Petunin inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6624.1.5 Mode, median and mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6624.1.6 Skewness and kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

24.2 Unimodal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6724.3 Other extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6724.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6824.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

25 Weak ordering 6925.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7025.2 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

25.2.1 Strict weak orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7025.2.2 Total preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7125.2.3 Ordered partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7125.2.4 Representation by functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

25.3 Related types of ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7225.4 All weak orders on a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

25.4.1 Combinatorial enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7225.4.2 Adjacency structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

25.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7425.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

26 Well-founded relation 7526.1 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7526.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7626.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7626.4 Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7726.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

27 Well-order 7827.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7827.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

27.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7927.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7927.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

27.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8027.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8027.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8127.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

28 Well-quasi-ordering 8228.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

CONTENTS vii

28.2 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8228.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8228.4 Wqo’s versus well partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8328.5 Infinite increasing subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8328.6 Properties of wqos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8328.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8428.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8428.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8428.10Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 85

28.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8528.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8728.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Chapter 1

Process calculus

In computer science, the process calculi (or process algebras) are a diverse family of related approaches for for-mally modelling concurrent systems. Process calculi provide a tool for the high-level description of interactions,communications, and synchronizations between a collection of independent agents or processes. They also providealgebraic laws that allow process descriptions to be manipulated and analyzed, and permit formal reasoning aboutequivalences between processes (e.g., using bisimulation). Leading examples of process calculi include CSP, CCS,ACP, and LOTOS.[1] More recent additions to the family include the π-calculus, the ambient calculus, PEPA, thefusion calculus and the join-calculus.

1.1 Essential features

While the variety of existing process calculi is very large (including variants that incorporate stochastic behaviour,timing information, and specializations for studying molecular interactions), there are several features that all processcalculi have in common:[2]

• Representing interactions between independent processes as communication (message-passing), rather than asmodification of shared variables.

• Describing processes and systems using a small collection of primitives, and operators for combining thoseprimitives.

• Defining algebraic laws for the process operators, which allow process expressions to be manipulated usingequational reasoning.

1.2 Mathematics of processes

To define a process calculus, one starts with a set of names (or channels) whose purpose is to provide means ofcommunication. In many implementations, channels have rich internal structure to improve efficiency, but this isabstracted away in most theoretic models. In addition to names, one needs a means to form new processes from old.The basic operators, always present in some form or other, allow:[3]

• parallel composition of processes

• specification of which channels to use for sending and receiving data

• sequentialization of interactions

• hiding of interaction points

• recursion or process replication

1

2 CHAPTER 1. PROCESS CALCULUS

1.2.1 Parallel composition

Parallel composition of two processes P and Q , usually written P |Q , is the key primitive distinguishing the processcalculi from sequential models of computation. Parallel composition allows computation in P and Q to proceedsimultaneously and independently. But it also allows interaction, that is synchronisation and flow of information fromP to Q (or vice versa) on a channel shared by both. Crucially, an agent or process can be connected to more than onechannel at a time.Channels may be synchronous or asynchronous. In the case of a synchronous channel, the agent sending a messagewaits until another agent has received the message. Asynchronous channels do not require any such synchronization.In some process calculi (notably the π-calculus) channels themselves can be sent in messages through (other) channels,allowing the topology of process interconnections to change. Some process calculi also allow channels to be createdduring the execution of a computation.

1.2.2 Communication

Interaction can be (but isn't always) a directed flow of information. That is, input and output can be distinguished asdual interaction primitives. Process calculi that make such distinctions typically define an input operator (e.g. x(v)) and an output operator (e.g. x⟨y⟩ ), both of which name an interaction point (here x ) that is used to synchronisewith a dual interaction primitive.Information should be exchanged, it will flow from the outputting to the inputting process. The output primitive willspecify the data to be sent. In x⟨y⟩ , this data is y . Similarly, if an input expects to receive data, one or more boundvariables will act as place-holders to be substituted by data, when it arrives. In x(v) , v plays that role. The choice ofthe kind of data that can be exchanged in an interaction is one of the key features that distinguishes different processcalculi.

1.2.3 Sequential composition

Sometimes interactions must be temporally ordered. For example, it might be desirable to specify algorithms such as:first receive some data on x and then send that data on y . Sequential composition can be used for such purposes. It iswell known from other models of computation. In process calculi, the sequentialisation operator is usually integratedwith input or output, or both. For example, the process x(v) · P will wait for an input on x . Only when this inputhas occurred will the process P be activated, with the received data through x substituted for identifier v .

1.2.4 Reduction semantics

The key operational reduction rule, containing the computational essence of process calculi, can be given solely interms of parallel composition, sequentialization, input, and output. The details of this reduction vary among thecalculi, but the essence remains roughly the same. The reduction rule is:

x⟨y⟩ · P | x(v) ·Q −→ P | Q[y/v]

The interpretation of this reduction rule is:

1. The process x⟨y⟩ ·P sends a message, here y , along the channel x . Dually, the process x(v) ·Q receives thatmessage on channel x .

2. Once the message has been sent, x⟨y⟩ · P becomes the process P , while x(v) ·Q becomes the process Q[y/v], which is Q with the place-holder v substituted by y , the data received on x .

The class of processes that P is allowed to range over as the continuation of the output operation substantially influ-ences the properties of the calculus.

1.3. DISCRETE AND CONTINUOUS PROCESS ALGEBRA 3

1.2.5 Hiding

Processes do not limit the number of connections that can be made at a given interaction point. But interaction pointsallow interference (i.e. interaction). For the synthesis of compact, minimal and compositional systems, the ability torestrict interference is crucial. Hiding operations allow control of the connections made between interaction pointswhen composing agents in parallel. Hiding can be denoted in a variety of ways. For example, in the π -calculus thehiding of a name x in P can be expressed as (ν x)P , while in CSP it might be written as P \ {x} .

1.2.6 Recursion and replication

The operations presented so far describe only finite interaction and are consequently insufficient for full computability,which includes non-terminating behaviour. Recursion and replication are operations that allow finite descriptionsof infinite behaviour. Recursion is well known from the sequential world. Replication !P can be understood asabbreviating the parallel composition of a countably infinite number of P processes:

!P = P |!P

1.2.7 Null process

Process calculi generally also include a null process (variously denoted as nil , 0 , STOP , δ , or some other appropriatesymbol) which has no interaction points. It is utterly inactive and its sole purpose is to act as the inductive anchor ontop of which more interesting processes can be generated.

1.3 Discrete and continuous process algebra

Process algebra has been studied for discrete time and continuous time (real time or dense time).[4]

1.4 History

In the first half of the 20th century, various formalisms were proposed to capture the informal concept of a com-putable function, with μ-recursive functions, Turing Machines and the lambda calculus possibly being the best-knownexamples today. The surprising fact that they are essentially equivalent, in the sense that they are all encodable intoeach other, supports the Church-Turing thesis. Another shared feature is more rarely commented on: they all aremost readily understood as models of sequential computation. The subsequent consolidation of computer science re-quired a more subtle formulation of the notion of computation, in particular explicit representations of concurrencyand communication. Models of concurrency such as the process calculi, Petri nets in 1962, and the Actor model in1973 emerged from this line of enquiry.Research on process calculi began in earnest with Robin Milner's seminal work on the Calculus of CommunicatingSystems (CCS) during the period from 1973 to 1980. C.A.R. Hoare's Communicating Sequential Processes (CSP)first appeared in 1978, and was subsequently developed into a full-fledged process calculus during the early 1980s.There was much cross-fertilization of ideas between CCS and CSP as they developed. In 1982 Jan Bergstra andJan Willem Klop began work on what came to be known as the Algebra of Communicating Processes (ACP), andintroduced the term process algebra to describe their work.[1] CCS, CSP, and ACP constitute the three major branchesof the process calculi family: the majority of the other process calculi can trace their roots to one of these three calculi.

1.5 Current research

Various process calculi have been studied and not all of them fit the paradigm sketched here. The most prominentexample may be the ambient calculus. This is to be expected as process calculi are an active field of study. Currentlyresearch on process calculi focuses on the following problems.

4 CHAPTER 1. PROCESS CALCULUS

• Developing new process calculi for better modeling of computational phenomena.

• Finding well-behaved subcalculi of a given process calculus. This is valuable because (1) most calculi are fairlywild in the sense that they are rather general and not much can be said about arbitrary processes; and (2)computational applications rarely exhaust the whole of a calculus. Rather they use only processes that are veryconstrained in form. Constraining the shape of processes is mostly studied by way of type systems.

• Logics for processes that allow one to reason about (essentially) arbitrary properties of processes, following theideas of Hoare logic.

• Behavioural theory: what does it mean for two processes to be the same? How can we decide whether twoprocesses are different or not? Can we find representatives for equivalence classes of processes? Generally,processes are considered to be the same if no context, that is other processes running in parallel, can detect adifference. Unfortunately, making this intuition precise is subtle and mostly yields unwieldy characterisations ofequality (which in most cases must also be undecidable, as a consequence of the halting problem). Bisimulationsare a technical tool that aids reasoning about process equivalences.

• Expressivity of calculi. Programming experience shows that certain problems are easier to solve in somelanguages than in others. This phenomenon calls for a more precise characterisation of the expressivity ofcalculi modeling computation than that afforded by the Church-Turing thesis. One way of doing this is toconsider encodings between two formalisms and see what properties encodings can potentially preserve. Themore properties can be preserved, the more expressive the target of the encoding is said to be. For processcalculi, the celebrated results are that the synchronous π -calculus is more expressive than its asynchronousvariant, has the same expressive power as the higher-order π -calculus, but is less than the ambient calculus.

• Using process calculus to model biological systems (stochasticπ -calculus, BioAmbients, Beta Binders, BioPEPA,Brane calculus). It is thought by some that the compositionality offered by process-theoretic tools can help bi-ologists to organise their knowledge more formally.

1.6 Software implementations

The ideas behind process algebra have given rise to several tools including:

• CADP

• Concurrency Workbench

• mCRL2 toolset

1.7 Relationship to other models of concurrency

The history monoid is the free object that is generically able to represent the histories of individual communicatingprocesses. A process calculus is then a formal language imposed on a history monoid in a consistent fashion.[5] Thatis, a history monoid can only record a sequence of events, with synchronization, but does not specify the allowedstate transitions. Thus, a process calculus is to a history monoid what a formal language is to a free monoid (a formallanguage is a subset of the set of all possible finite-length strings of an alphabet generated by the Kleene star).The use of channels for communication is one of the features distinguishing the process calculi from other models ofconcurrency, such as Petri nets and the Actor model (see Actor model and process calculi). One of the fundamentalmotivations for including channels in the process calculi was to enable certain algebraic techniques, thereby makingit easier to reason about processes algebraically.

1.8 See also• Stochastic probe

1.9. REFERENCES 5

1.9 References[1] Baeten, J.C.M. (2004). “A brief history of process algebra” (PDF). Rapport CSR 04-02 (Vakgroep Informatica, Technische

Universiteit Eindhoven).

[2] Pierce, Benjamin. “Foundational Calculi for Programming Languages”. The Computer Science and Engineering Handbook.CRC Press. pp. 2190–2207. ISBN 0-8493-2909-4.

[3] Baeten, J.C.M.; Bravetti, M. (August 2005). “A Generic Process Algebra”. Algebraic Process Calculi: The First TwentyFive Years and Beyond (BRICS Notes Series NS-05-3). Bertinoro, Forl`ı, Italy: BRICS, Department of Computer Science,University of Aarhus. Retrieved 2007-12-29.

[4] Baeten, J. C. M.; Middelburg, C. A. “Process algebra with timing: Real time and discrete time”. CiteSeerX: 10 .1 .1 .42 .729.

[5] Mazurkiewicz, Antoni (1995). “Introduction to Trace Theory”. In Diekert, V.; Rozenberg, G. The Book of Traces(POSTSCRIPT). Singapore: World Scientific. pp. 3–41. ISBN 981-02-2058-8.

1.10 Further reading• Matthew Hennessy: Algebraic Theory of Processes, The MIT Press, ISBN 0-262-08171-7.

• C. A. R. Hoare: Communicating Sequential Processes, Prentice Hall, ISBN 0-13-153289-8.

• This book has been updated by Jim Davies at the Oxford University Computing Laboratory and the newedition is available for download as a PDF file at the Using CSP website.

• Robin Milner: A Calculus of Communicating Systems, Springer Verlag, ISBN 0-387-10235-3.

• Robin Milner: Communicating and Mobile Systems: the Pi-Calculus, Springer Verlag, ISBN 0-521-65869-1.

• Andrew Mironov: Theory of processes

Chapter 2

Quasitransitive relation

Quasitransitivity is a weakened version of transitivity that is used in social choice theory or microeconomics. In-formally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere. The concept wasintroduced by Sen (1969) to study the consequences of Arrow’s theorem.

2.1 Formal definition

A binary relation T over a set X is quasitransitive if for all a, b, and c in X the following holds:

(aT b) ∧ ¬(bT a) ∧ (bT c) ∧ ¬(cT b) ⇒ (aT c) ∧ ¬(cT a).

If the relation is also antisymmetric, T is transitive.Alternately, for a relation T, define the asymmetric or “strict” part P:

(aP b) ⇔ (aT b) ∧ ¬(bT a).

Then T is quasitransitive iff P is transitive.

2.2 Examples

Preferences are assumed to be quasitransitive (rather than transitive) in some economic contexts. The classic exampleis a person indifferent between 10 and 11 grams of sugar and indifferent between 11 and 12 grams of sugar, but whoprefers 12 grams of sugar to 10. Similarly, the Sorites paradox can be resolved by weakening assumed transitivity ofcertain relations to quasitransitivity.

2.3 Properties

• Every transitive relation is quasitransitive; every quasitransitive relation is an acyclic relation. In each case theconverse does not hold in general.

2.4 See also

• Intransitivity

• Reflexive relation

6

2.5. REFERENCES 7

2.5 References• Bossert, Walter; Suzumura, Kōtarō (2010). Consistency, choice and rationality. Harvard University Press.

ISBN 0674052994.

• Sen, A. (1969). “Quasi-transitivity, rational choice and collective decisions”. Rev. Econ. Stud. 36: 381–393.doi:10.2307/2296434. Zbl 0181.47302.

Chapter 3

Quotient by an equivalence relation

This article is about a generalization to category theory, used in scheme theory. For the common meaning, seeEquivalence class.

In mathematics, given a category C, a quotient of an object X by an equivalence relation f : R → X × X is acoequalizer for the pair of maps

Rf→X ×X

pri→X, i = 1, 2,

where R is an object in C and "f is an equivalence relation” means that, for any object T in C, the image (which isa set) of f : R(T ) = Mor(T,R) → X(T ) × X(T ) is an equivalence relation; that is, (x, y) is in it if and only if(y, x) is in it, etc.The basic case in practice is when C is the category of all schemes over some scheme S. But the notion is flexible andone can also take C to be the category of sheaves.

3.1 Examples

• Let X be a set and consider some equivalence relation on it. Let Q be the set of all equivalence classes in X.Then the map q : X → Q that sends an element x to an equivalence class to which x belong is a quotient.

• In the above example, Q is a subset of the power set H of X. In algebraic geometry, one might replace Hby a Hilbert scheme or disjoint union of Hilbert schemes. In fact, Grothendieck constructed a relative Picardscheme of a flat projective schemeX[1] as a quotientQ (of the scheme Z parametrizing relative effective divisorson X) that is a closed scheme of a Hilbert scheme H. The quotient map q : Z → Q can then be thought of asa relative version of the Abel map.

3.2 See also

• categorical quotient, a special case

3.3 Notes

[1] One also needs to assume the geometric fibers are integral schemes; Mumford’s example shows the “integral” cannot beomitted.

8

3.4. REFERENCES 9

3.4 References• Nitsure, N. Construction of Hilbert and Quot schemes. Fundamental algebraic geometry: Grothendieck’s FGA

explained, Mathematical Surveys and Monographs 123, American Mathematical Society 2005, 105–137.

Chapter 4

Rational consequence relation

In logic, a rational consequence relation is a non-monotonic consequence relation satisfying certain properties listedbelow.

4.1 Properties

A rational consequence relation satisfies:

REF Reflexivity θ ⊢ θ

and the so-called Gabbay-Makinson rules:

LLE Left Logical Equivalence θ⊢ψ θ≡ϕϕ⊢ψ

RWE Right-hand weakening θ⊢ϕ ϕ|=ψθ⊢ψ

CMO Cautious monotonicity θ⊢ϕ θ⊢ψθ∧ψ⊢ϕ

DIS Logical or (ie disjunction) on left hand side θ⊢ψ ϕ⊢ψθ∨ϕ⊢ψ

AND Logical and on right hand side θ⊢ϕ θ⊢ψθ⊢ϕ∧ψ

RMO Rational monotonicity ϕ̸⊢¬θ ϕ⊢ψϕ∧θ⊢ψ

4.2 Uses

The rational consequence relation is non-monotonic, and the relation θ ⊢ ϕ is intended to carry the meaning thetausually implies phi or phi usually follows from theta. In this sense it is more useful for modeling some everydaysituations than a monotone consequence relation because the latter relation models facts in a more strict booleanfashion - something either follows under all circumstances or it does not.

4.2.1 Example

The statement “If a cake contains sugar then it tastes good” implies under a monotone consequence relation the state-ment “If a cake contains sugar and soap then it tastes good.” Clearly this doesn't match our own understanding ofcakes. By asserting “If a cake contains sugar then it usually tastes good” a rational consequence relation allows fora more realistic model of the real world, and certainly it does not automatically follow that “If a cake contains sugarand soap then it usually tastes good.”

Note that if we also have the information “If a cake contains sugar then it usually contains butter” then we may legallyconclude (under CMO) that “If a cake contains sugar and butter then it usually tastes good.”. Equally in the absence

10

4.3. CONSEQUENCES 11

of a statement such as “If a cake contains sugar then usually it contains no soap" then we may legally conclude fromRMO that “If the cake contains sugar and soap then it usually tastes good.”If this latter conclusion seems ridiculous to you then it is likely that you are subconsciously asserting your own pre-conceived knowledge about cakes when evaluating the validity of the statement. That is, from your experience youknow that cakes which contain soap are likely to taste bad so you add to the system your own knowledge such as“Cakes which contain sugar do not usually contain soap.”, even though this knowledge is absent from it. If the conclu-sion seems silly to you then you might consider replacing the word soap with the word eggs to see if it changes yourfeelings.

4.2.2 Example

Consider the sentences:

• Young people are usually happy

• Drug abusers are usually not happy

• Drug abusers are usually young

We may consider it reasonable to conclude:

• Young drug abusers are usually not happy

This would not be a valid conclusion under a monotonic deduction system (omitting of course the word 'usually'),since the third sentence would contradict the first two. In contrast the conclusion follows immediately using theGabbay-Makinson rules: applying the rule CMO to the last two sentences yields the result.

4.3 Consequences

The following consequences follow from the above rules:

MP Modus ponens θ⊢ϕ θ⊢(ϕ→ψ)θ⊢ψ

MP is proved via the rules AND and RWE.

CON Conditionalisation θ∧ϕ⊢ψθ⊢(ϕ→ψ)

CC Cautious Cut θ⊢ϕ θ∧ϕ⊢ψθ⊢ψ

The notion of Cautious Cut simply encapsulates the operation of conditionalisation, followed by MP.It may seem redundant in this sense, but it is often used in proofs so it is useful to have a name forit to act as a shortcut.

SCL Supraclassity θ|=ϕθ⊢ϕ

SCL is proved trivially via REF and RWE.

4.4 Rational consequence relations via atom preferences

Let L = {p1, . . . , pn} be a finite language. An atom is a formula of the form∧ni=1 p

ϵi (where p1 = p and p−1 = ¬p

). Notice that there is a unique valuation which makes any given atom true (and conversely each valuation satisfiesprecisely one atom). Thus an atom can be used to represent a preference about what we believe ought to be true.Let AtL be the set of all atoms in L. For θ ∈ SL, define Sθ = {α ∈ AtL|α |=SC θ} .Let s⃗ = s1, . . . , sm be a sequence of subsets of AtL . For θ , ϕ in SL, let the relation ⊢s⃗ be such that θ ⊢s⃗ ϕ if oneof the following holds:

12 CHAPTER 4. RATIONAL CONSEQUENCE RELATION

1. Sθ ∩ si = ∅ for each 1 ≤ i ≤ m

2. Sθ ∩ si ̸= ∅ for some 1 ≤ i ≤ m and for the least such i, Sθ ∩ si ⊆ Sϕ .

Then the relation ⊢s⃗ is a rational consequence relation. This may easily be verified by checking directly that it satisfiesthe GM-conditions.The idea behind the sequence of atom sets is that the earlier sets account for the most likely situations such as “youngpeople are usually law abiding” whereas the later sets account for the less likely situations such as “young joyridersare usually not law abiding”.

4.4.1 Notes

1. By the definition of the relation ⊢s⃗ , the relation is unchanged if we replace s2 with s2 \s1 , s3 with s3 \s2 \s1... and sm with sm \

∪m−1i=1 si . In this way we make each si disjoint. Conversely it makes no difference to the

rcr ⊢s⃗ if we add to subsequent si atoms from any of the preceding si .

4.5 The representation theorem

It can be proven that any rational consequence relation on a finite language is representable via a sequence of atompreferences above. That is, for any such rational consequence relation ⊢ there is a sequence s⃗ = s1, . . . , sm of subsetsof AtL such that the associated rcr ⊢s⃗ is the same relation: ⊢s⃗=⊢

4.5.1 Notes

1. By the above property of ⊢s⃗ , the representation of an rcr ⊢ need not be unique - if the si are not disjoint thenthey can be made so without changing the rcr and conversely if they are disjoint then each subsequent set cancontain any of the atoms of the previous sets without changing the rcr.

4.6 References• A mathematical paper in which the GM rules are defined

Chapter 5

Reduct

This article is about a relation on algebraic structures. For reducts in abstract rewriting, see Confluence (abstractrewriting).

In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of theoperations and relations of that structure. The converse of “reduct” is “expansion.”

5.1 Definition

Let A be an algebraic structure (in the sense of universal algebra) or equivalently a structure in the sense of modeltheory, organized as a set X together with an indexed family of operations and relations φᵢ on that set, with index setI. Then the reduct of A defined by a subset J of I is the structure consisting of the set X and J-indexed family ofoperations and relations whose j-th operation or relation for j∈J is the j-th operation or relation of A. That is, thisreduct is the structure A with the omission of those operations and relations φi for which i is not in J.A structure A is an expansion of B just when B is a reduct of A. That is, reduct and expansion are mutual converses.

5.2 Examples

The monoid (Z, +, 0) of integers under addition is a reduct of the group (Z, +, −, 0) of integers under addition andnegation, obtained by omitting negation. By contrast, the monoid (N,+,0) of natural numbers under addition is notthe reduct of any group.Conversely the group (Z, +, −, 0) is the expansion of the monoid (Z, +, 0), expanding it with the operation of negation.

5.3 References• Burris, Stanley N.; H. P. Sankappanavar (1981). ACourse in Universal Algebra. Springer. ISBN 3-540-90578-

2.

• Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 0-521-30442-3.

13

Chapter 6

Reflexive closure

In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X thatcontains R.For example, if X is a set of distinct numbers and x R y means "x is less than y", then the reflexive closure of R is therelation "x is less than or equal to y".

6.1 Definition

The reflexive closure S of a relation R on a set X is given by

S = R ∪ {(x, x) : x ∈ X}

In words, the reflexive closure of R is the union of R with the identity relation on X.

6.2 See also• Transitive closure

• Symmetric closure

6.3 References• Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8

14

Chapter 7

Reflexive relation

In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself. In otherwords, a relation ~ on a set S is reflexive when x ~ x holds true for every x in S, formally: when ∀x∈S: x~x holds.[1][2]

An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number isequal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity.

7.1 Related terms

A relation that is irreflexive, or anti-reflexive, is a binary relation on a set where no element is related to itself. Anexample is the “greater than” relation (x>y) on the real numbers. Note that not every relation which is not reflexiveis irreflexive; it is possible to define relations where some elements are related to themselves but others are not (i.e.,neither all nor none are). For example, the binary relation “the product of x and y is even” is reflexive on the set ofeven numbers, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of natural numbers.A relation ~ on a set S is called quasi-reflexive if every element that is related to some element is also related to itself,formally: if ∀x,y∈S: x~y ⇒ x~x ∧ y~y. An example is the relation “has the same limit as” on the set of sequences ofreal numbers: not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the samelimit as some sequence, then it has the same limit as itself.The reflexive closure ≃ of a binary relation ~ on a set S is the smallest reflexive relation on S that is a superset of ~.Equivalently, it is the union of ~ and the identity relation on S, formally: (≃) = (~) ∪ (=). For example, the reflexiveclosure of x<y is x≤y.The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set S is the smallest relation ≆ such that ≆shares the same reflexive closure as ~. It can be seen in a way as the opposite of the reflexive closure. It is equivalentto the complement of the identity relation on S with regard to ~, formally: (≆) = (~) \ (=). That is, it is equivalent to~ except for where x~x is true. For example, the reflexive reduction of x≤y is x<y.

7.2 Examples

Examples of reflexive relations include:

• “is equal to” (equality)

• “is a subset of” (set inclusion)

• “divides” (divisibility)

• “is greater than or equal to”

• “is less than or equal to”

Examples of irreflexive relations include:

15

16 CHAPTER 7. REFLEXIVE RELATION

• “is not equal to”

• “is coprime to” (for the integers>1, since 1 is coprime to itself)

• “is a proper subset of”

• “is greater than”

• “is less than”

7.3 Number of reflexive relations

The number of reflexive relations on an n-element set is 2n2−n.[3]

7.4. PHILOSOPHICAL LOGIC 17

7.4 Philosophical logic

Authors in philosophical logic often use deviating designations. A reflexive and a quasi-reflexive relation in themathematical sense is called a totally reflexive and a reflexive relation in philosophical logic sense, respectively.[4][5]

7.5 See also

• Binary relation

• Symmetric relation

• Transitive relation

18 CHAPTER 7. REFLEXIVE RELATION

• Coreflexive relation

7.6 Notes[1] Levy 1979:74

[2] Relational Mathematics, 2010

[3] On-Line Encyclopedia of Integer Sequences A053763

[4] Alan Hausman, Howard Kahane, Paul Tidman (2013). Logic and Philosophy—AModern Introduction. Wadsworth. ISBN1-133-05000-X. Here: p.327-328

[5] D.S. Clarke, Richard Behling (1998). Deductive Logic — An Introduction to Evaluation Techniques and Logical Theory.University Press of America. ISBN 0-7618-0922-8. Here: p.187

7.7 References• Levy, A. (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002,

Dover. ISBN 0-486-42079-5

• Lidl, R. and Pilz, G. (1998). Applied abstract algebra, Undergraduate Texts in Mathematics, Springer-Verlag.ISBN 0-387-98290-6

• Quine, W. V. (1951). Mathematical Logic, Revised Edition. Reprinted 2003, Harvard University Press. ISBN0-674-55451-5

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

7.8 External links• Hazewinkel, Michiel, ed. (2001), “Reflexivity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

Chapter 8

Relation algebra

Not to be confused with relational algebra, a framework for finitary relations and relational databases.

In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involutioncalled converse, a unary operation. The motivating example of a relation algebra is the algebra 2X² of all binaryrelations on a set X, that is, subsets of the cartesian square X2, with R•S interpreted as the usual composition ofbinary relations R and S, and with the converse of R interpreted as the inverse relation.Relation algebra emerged in the 19th-century work of Augustus De Morgan and Charles Peirce, which culminatedin the algebraic logic of Ernst Schröder. The equational form of relation algebra treated here was developed byAlfred Tarski and his students, starting in the 1940s. Tarski and Givant (1987) applied relation algebra to a variable-free treatment of axiomatic set theory, with the implication that mathematics founded on set theory could itself beconducted without variables.

8.1 Definition

A relation algebra (L, ∧, ∨, −, 0, 1, •, I, ˘ ) is an algebraic structure equipped with the Boolean operations ofconjunction x∧y, disjunction x∨y, and negation x−, the Boolean constants 0 and 1, the relational operations of com-position x•y and converse x ˘ , and the relational constant I, such that these operations and constants satisfy certainequations constituting an axiomatization of relation algebras. A relation algebra is to a system of binary relationson a set containing the empty (0), complete (1), and identity (I) relations and closed under these five operations as agroup is to a system of permutations of a set containing the identity permutation and closed under composition andinverse.Following Jónsson and Tsinakis (1993) it is convenient to define additional operations x◁y = x•y˘ , and, dually, x▷y= x ˘ •y . Jónsson and Tsinakis showed that I◁x = x▷I, and that both were equal to x ˘ . Hence a relation algebra canequally well be defined as an algebraic structure (L, ∧, ∨, −, 0, 1, •, I, ◁, ▷). The advantage of this signature overthe usual one is that a relation algebra can then be defined in full simply as a residuated Boolean algebra for whichI◁x is an involution, that is, I◁(I◁x) = x . The latter condition can be thought of as the relational counterpart of theequation 1/(1/x) = x for ordinary arithmetic reciprocal, and some authors use reciprocal as a synonym for converse.Since residuated Boolean algebras are axiomatized with finitely many identities, so are relation algebras. Hence thelatter form a variety, the variety RA of relation algebras. Expanding the above definition as equations yields thefollowing finite axiomatization.

8.1.1 Axioms

The axioms B1-B10 below are adapted from Givant (2006: 283), and were first set out by Tarski in 1948.[1]

L is a Boolean algebra under binary disjunction, ∨, and unary complementation ()−:

B1: A ∨ B = B ∨ A

B2: A ∨ (B ∨ C) = (A ∨ B) ∨ C

19

20 CHAPTER 8. RELATION ALGEBRA

B3: (A− ∨ B)− ∨ (A− ∨ B−)− = A

This axiomatization of Boolean algebra is due to Huntington (1933). Note that the meet of the implied Booleanalgebra is not the • operator (even though it distributes over ∨ like a meet does), nor is the 1 of the Boolean algebrathe I constant.L is a monoid under binary composition (•) and nullary identity I:

B4: A•(B•C) = (A•B)•CB5: A•I = A

Unary converse () ˘ is an involution with respect to composition:

B6: A ˘̆ = A

B7: (A•B) ˘ = B ˘ •A ˘

Axiom B6 defines conversion as an involution, whereas B7 expresses the antidistributive property of conversionrelative to composition.[2]

Converse and composition distribute over disjunction:

B8: (A∨B) ˘ = A ˘ ∨B ˘

B9: (A∨B)•C = (A•C)∨(B•C)

B10 is Tarski’s equational form of the fact, discovered by Augustus De Morgan, that A•B ≤ C− ↔ A ˘ •C ≤ B− ↔C•B ˘ ≤ A−.

B10: (A ˘ •(A•B)−)∨B− = B−

These axioms are ZFC theorems; for the purely BooleanB1-B3, this fact is trivial. After each of the following axiomsis shown the number of the corresponding theorem in chpt. 3 of Suppes (1960), an exposition of ZFC: B4 27, B545, B6 14, B7 26, B8 16, B9 23.

8.2 Expressing properties of binary relations in RA

The following table shows how many of the usual properties of binary relations can be expressed as succinct RAequalities or inequalities. Below, an inequality of the form A≤B is shorthand for the Boolean equation A∨B = B.The most complete set of results of this nature is chpt. C of Carnap (1958), where the notation is rather distant fromthat of this entry. Chpt. 3.2 of Suppes (1960) contains fewer results, presented as ZFC theorems and using a notationthat more resembles that of this entry. Neither Carnap nor Suppes formulated their results using the RA of this entry,or in an equational manner.

8.3 Expressive power

The metamathematics of RA are discussed at length in Tarski and Givant (1987), and more briefly in Givant (2006).RA consists entirely of equations manipulated using nothing more than uniform replacement and the substitution ofequals for equals. Both rules are wholly familiar from school mathematics and from abstract algebra generally. HenceRA proofs are carried out in a manner familiar to all mathematicians, unlike the case in mathematical logic generally.RA can express any (and up to logical equivalence, exactly the) first-order logic (FOL) formulas containing no morethan three variables. (A given variable can be quantified multiple times and hence quantifiers can be nested arbitrarilydeeply by “reusing” variables.) Surprisingly, this fragment of FOL suffices to express Peano arithmetic and almostall axiomatic set theories ever proposed. Hence RA is, in effect, a way of algebraizing nearly all mathematics,

8.4. EXAMPLES 21

while dispensing with FOL and its connectives, quantifiers, turnstiles, and modus ponens. Because RA can expressPeano arithmetic and set theory, Gödel’s incompleteness theorems apply to it; RA is incomplete, incompletable, andundecidable. (N.B. The Boolean algebra fragment of RA is complete and decidable.)The representable relation algebras, forming the class RRA, are those relation algebras isomorphic to some re-lation algebra consisting of binary relations on some set, and closed under the intended interpretation of the RAoperations. It is easily shown, e.g. using the method of pseudoelementary classes, that RRA is a quasivariety, thatis, axiomatizable by a universal Horn theory. In 1950, Roger Lyndon proved the existence of equations holding inRRA that did not hold in RA. Hence the variety generated by RRA is a proper subvariety of the variety RA. In1955, Alfred Tarski showed that RRA is itself a variety. In 1964, Donald Monk showed that RRA has no finiteaxiomatization, unlike RA which is finitely axiomatized by definition.

8.3.1 Q-Relation Algebras

An RA is a Q-Relation Algebra (QRA) if, in addition to B1-B10, there exist some A and B such that (Tarski andGivant 1987: §8.4):

Q0: A ˘ •A ≤ I

Q1: B ˘ •B ≤ I

Q2: A ˘ •B = 1

Essentially these axioms imply that the universe has a (non-surjective) pairing relation whose projections are A andB. It is a theorem that every QRA is a RRA (Proof by Maddux, see Tarski & Givant 1987: 8.4(iii) ).Every QRA is representable (Tarski and Givant 1987). That not every relation algebra is representable is a fun-damental way RA differs from QRA and Boolean algebras which, by Stone’s representation theorem for Booleanalgebras, are always representable as sets of subsets of some set, closed under union, intersection, and complement.

8.4 Examples

1. Any Boolean algebra can be turned into a RA by interpreting conjunction as composition (the monoid multipli-cation •), i.e. x•y is defined as x∧y. This interpretation requires that converse interpret identity (ў = y), and that bothresiduals y\x and x/y interpret the conditional y→x (i.e., ¬y∨x).2. The motivating example of a relation algebra depends on the definition of a binary relation R on a set X as anysubset R ⊆ X², where X² is the Cartesian square of X. The power set 2X² consisting of all binary relations on X isa Boolean algebra. While 2X² can be made a relation algebra by taking R•S = R∧S, as per example (1) above, thestandard interpretation of • is instead x(R•S)z = ∃y.xRySz. That is, the ordered pair (x,z) belongs to the relation R•Sjust when there exists y ∈ X such that (x,y) ∈ R and (y,z) ∈ S. This interpretation uniquely determines R\S as consistingof all pairs (y,z) such that for all x ∈ X, if xRy then xSz. Dually, S/R consists of all pairs (x,y) such that for all z ∈ X, ifyRz then xSz. The translation ў = ¬(y\¬I) then establishes the converse R ˘ of R as consisting of all pairs (y,x) suchthat (x,y) ∈ R.3. An important generalization of the previous example is the power set 2E where E ⊆X² is any equivalence relation onthe set X. This is a generalization because X² is itself an equivalence relation, namely the complete relation consistingof all pairs. While 2E is not a subalgebra of 2X² when E ≠ X² (since in that case it does not contain the relation X²,the top element 1 being E instead of X²), it is nevertheless turned into a relation algebra using the same definitionsof the operations. Its importance resides in the definition of a representable relation algebra as any relation algebraisomorphic to a subalgebra of the relation algebra 2E for some equivalence relation E on some set. The previoussection says more about the relevant metamathematics.4. If group sum or product interprets composition, group inverse interprets converse, group identity interprets I, andif R is a one to one correspondence, so that R ˘ •R = R•R

˘ = I,[3] then L is a group as well as a monoid. B4-B7become well-known theorems of group theory, so that RA becomes a proper extension of group theory as well as ofBoolean algebra.

22 CHAPTER 8. RELATION ALGEBRA

8.5 Historical remarks

DeMorgan founded RA in 1860, but C. S. Peirce took it much further and became fascinated with its philosophicalpower. The work of DeMorgan and Peirce came to be known mainly in the extended and definitive form ErnstSchröder gave it in Vol. 3 of his Vorlesungen (1890–1905). Principia Mathematica drew strongly on Schröder’s RA,but acknowledged him only as the inventor of the notation. In 1912, Alwin Korselt proved that a particular formulain which the quantifiers were nested four deep had no RA equivalent.[4] This fact led to a loss of interest in RA untilTarski (1941) began writing about it. His students have continued to develop RA down to the present day. Tarskireturned to RA in the 1970s with the help of Steven Givant; this collaboration resulted in the monograph by Tarskiand Givant (1987), the definitive reference for this subject. For more on the history ofRA, see Maddux (1991, 2006).

8.6 Software

• RelMICS / Relational Methods in Computer Science maintained by Wolfram Kahl

• Carsten Sinz: ARA / An Automatic Theorem Prover for Relation Algebras

8.7 See also

8.8 Footnotes[1] Alfred Tarski (1948) “Abstract: Representation Problems for Relation Algebras,” Bulletin of the AMS 54: 80.

[2] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer. pp. 4 and 8.ISBN 978-3-211-82971-4.

[3] Tarski, A. (1941), p. 87.

[4] Korselt did not publish his finding. It was first published in Leopold Loewenheim (1915) "Über Möglichkeiten im Rela-tivkalkül,” Mathematische Annalen 76: 447–470. Translated as “On possibilities in the calculus of relatives” in Jean vanHeijenoort, 1967. A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press: 228–251.

8.9 References

• Rudolf Carnap (1958) Introduction to Symbolic Logic and its Applications. Dover Publications.

• Givant, Steven (2006). “The calculus of relations as a foundation for mathematics”. Journal of AutomatedReasoning 37: 277–322. doi:10.1007/s10817-006-9062-x.

• Halmos, P. R., 1960. Naive Set Theory. Van Nostrand.

• Leon Henkin, Alfred Tarski, and Monk, J. D., 1971. Cylindric Algebras, Part 1, and 1985, Part 2. NorthHolland.

• Hirsch R., and Hodkinson, I., 2002, Relation Algebra byGames, vol. 147 in Studies in Logic and the Foundationsof Mathematics. Elsevier Science.

• Jónsson, Bjarni; Tsinakis, Constantine (1993). “Relation algebras as residuated Boolean algebras”. AlgebraUniversalis 30: 469–78. doi:10.1007/BF01195378.

• Maddux, Roger (1991). “The Origin of Relation Algebras in the Development and Axiomatization of theCalculus of Relations” (PDF). Studia Logica 50 (3–4): 421–455. doi:10.1007/BF00370681.

• --------, 2006. Relation Algebras, vol. 150 in Studies in Logic and the Foundations of Mathematics. ElsevierScience.

• Patrick Suppes, 1960. Axiomatic Set Theory. Van Nostrand. Dover reprint, 1972. Chpt. 3.

8.10. EXTERNAL LINKS 23

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press.

• Tarski, Alfred (1941). “On the calculus of relations”. Journal of Symbolic Logic 6: 73–89. doi:10.2307/2268577.

• ------, and Givant, Steven, 1987. A Formalization of Set Theory without Variables. Providence RI: AmericanMathematical Society.

8.10 External links• Yohji AKAMA, Yasuo Kawahara, and Hitoshi Furusawa, "Constructing Allegory from Relation Algebra and

Representation Theorems."

• Richard Bird, Oege de Moor, Paul Hoogendijk, "Generic Programming with Relations and Functors."

• R.P. de Freitas and Viana, "A Completeness Result for Relation Algebra with Binders."

• Peter Jipsen:

• Relation algebras. In Mathematical structures. If there are problems with LaTeX, see an old HTMLversion here.

• "Foundations of Relations and Kleene Algebra."• "Computer Aided Investigations of Relation Algebras."• "A Gentzen System And Decidability For Residuated Lattices.”

• Vaughan Pratt:

• "Origins of the Calculus of Binary Relations." A historical treatment.• "The Second Calculus of Binary Relations."

• Priss, Uta:

• "An FCA interpretation of Relation Algebra."• "Relation Algebra and FCA" Links to publications and software

• Kahl, Wolfram, and Schmidt, Gunther, "Exploring (Finite) Relation Algebras Using Tools Written in Haskell."See homepage of the whole project.

Chapter 9

Relation construction

In logic and mathematics, relation construction and relational constructibility have to do with the ways that onerelation is determined by an indexed family or a sequence of other relations, called the relation dataset. The relationin the focus of consideration is called the faciendum. The relation dataset typically consists of a specified relationover sets of relations, called the constructor, the factor, or the method of construction, plus a specified set of otherrelations, called the faciens, the ingredients, or the makings.Relation composition and relation reduction are special cases of relation constructions.

9.1 See also• Projection

• Relation

• Relation composition

• Relation reduction

24

Chapter 10

Representation (mathematics)

In mathematics, representation is a very general relationship that expresses similarities between objects. Roughlyspeaking, a collection Y of mathematical objects may be said to represent another collection X of objects, providedthat the properties and relationships existing among the representing objects yi conform in some consistent way tothose existing among the corresponding represented objects xi. Somewhat more formally, for a set Π of propertiesand relations, a Π-representation of some structure X is a structure Y that is the image of X under a homomorphismthat preserves Π. The label representation is sometimes also applied to the homomorphism itself.

10.1 Representation theory

Perhaps the most well-developed example of this general notion is the subfield of abstract algebra called representationtheory, which studies the representing of elements of algebraic structures by linear transformations of vector spaces.

10.2 Other examples

Although the term representation theory is well established in the algebraic sense discussed above, there are manyother uses of the term representation throughout mathematics.

10.2.1 Graph theory

An active area of graph theory is the exploration of isomorphisms between graphs and other structures. A keyclass of such problems stems from the fact that, like adjacency in undirected graphs, intersection of sets (or, moreprecisely, non-disjointness) is a symmetric relation. This gives rise to the study of intersection graphs for innumerablefamilies of sets.[1] One foundational result here, due to Paul Erdős and colleagues, is that every n-vertex graph maybe represented in terms of intersection among subsets of a set of size no more than n2/4.[2]

Representing a graph by such algebraic structures as its adjacency matrix and Laplacian matrix gives rise to the fieldof spectral graph theory.[3]

10.2.2 Order theory

Dual to the observation above that every graph is an intersection graph is the fact that every partially ordered set isisomorphic to a collection of sets ordered by the containment (or inclusion) relation ⊆. Among the posets that ariseas the containment orders for natural classes of objects are the Boolean lattices and the orders of dimension n.[4]

Many partial orders arise from (and thus can be represented by) collections of geometric objects. Among them arethe n-ball orders. The 1-ball orders are the interval-containment orders, and the 2-ball orders are the so-called circleorders, the posets representable in terms of containment among disks in the plane. A particularly nice result in thisfield is the characterization of the planar graphs as those graphs whose vertex-edge incidence relations are circleorders.[5]

25

26 CHAPTER 10. REPRESENTATION (MATHEMATICS)

There are also geometric representations that are not based on containment. Indeed, one of the best studied classesamong these are the interval orders,[6] which represent the partial order in terms of what might be called disjointprecedence of intervals on the real line: each element x of the poset is represented by an interval [x1, x2] such that forany y and z in the poset, y is below z if and only if y2 < z1.

10.2.3 Polysemy

Under certain circumstances, a single function f:X → Y is at once an isomorphism from several mathematical struc-tures on X. Since each of those structures may be thought of, intuitively, as a meaning of the image Y—one of thethings that Y is trying to tell us—this phenomenon is called polysemy, a term borrowed from linguistics. Examplesinclude:

• intersection polysemy—pairs of graphs G1 and G2 on a common vertex set V that can be simultaneouslyrepresented by a single collection of sets Sv such that any distinct vertices u and w in V...

are adjacent in G1 if and only if their corresponding sets intersect ( Su ∩ Sw ≠ Ø ), andare adjacent in G2 if and only if the complements do ( SuC ∩ SwC ≠ Ø ).[7]

• competition polysemy—motivated by the study of ecological food webs, in which pairs of species may haveprey in common or have predators in common. A pair of graphs G1 and G2 on one vertex set is competitionpolysemic if and only if there exists a single directed graph D on the same vertex set such that any distinctvertices u and v...

are adjacent in G1 if and only if there is a vertex w such that both uw and vw are arcs in D,andare adjacent in G2 if and only if there is a vertex w such that both wu and wv are arcs in D.[8]

• interval polysemy—pairs of posets P1 and P2 on a common ground set that can be simultaneously representedby a single collection of real intervals that is an interval-order representation of P1 and an interval-containmentrepresentation of P2.[9]

10.3 See also• Representation theorems

• Model theory

10.4 References[1] • McKee, Terry A.; McMorris, F. R. (1999), Topics in Intersection Graph Theory, SIAM Monographs on Discrete

Mathematics and Applications, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 0-89871-430-3, MR 1672910

[2] Erdős, Paul; Goodman, A. W.; Pósa, Louis (1966), “The representation of a graph by set intersections”, Canadian Journalof Mathematics 18 (1): 106–112, doi:10.4153/cjm-1966-014-3, MR 0186575

[3] • Biggs, Norman (1994), Algebraic Graph Theory, Cambridge Mathematical Library, Cambridge University Press,ISBN 978-0-521-45897-9, MR 1271140

[4] • Trotter, William T. (1992), Combinatorics and Partially Ordered Sets: Dimension Theory, Johns Hopkins Series in theMathematical Sciences, Baltimore: The Johns Hopkins University Press, ISBN 978-0-8018-4425-6, MR 1169299

[5] • Scheinerman, Edward (1991), “A note on planar graphs and circle orders”, SIAM Journal on Discrete Mathematics 4(3): 448–451, doi:10.1137/0404040, MR 1105950

[6] • Fishburn, Peter C. (1985), Interval Orders and Interval Graphs: A Study of Partially Ordered Sets, Wiley-InterscienceSeries in Discrete Mathematics, John Wiley & Sons, ISBN 978-0-471-81284-5, MR 0776781

10.4. REFERENCES 27

[7] • Tanenbaum, Paul J. (1999), “Simultaneous intersection representation of pairs of graphs”, Journal of Graph Theory32 (2): 171–190, doi:10.1002/(SICI)1097-0118(199910)32:2<171::AID-JGT7>3.0.CO;2-N, MR 1709659

[8] • Fischermann, Miranca; Knoben, Werner; Kremer, Dirk; Rautenbachh, Dieter (2004), “Competition polysemy”,Discrete Mathematics 282 (1–3): 251–255, doi:10.1016/j.disc.2003.11.014, MR 2059526

[9] • Tanenbaum, Paul J. (1996), “Simultaneous representation of interval and interval-containment orders”, Order 13 (4):339–350, doi:10.1007/BF00405593, MR 1452517

Chapter 11

Separoid

In mathematics, a separoid is a binary relation between disjoint sets which is stable as an ideal in the canonical orderinduced by inclusion. Many mathematical objects which appear to be quite different, find a common generalisation inthe framework of separoids; e.g., graphs, configurations of convex sets, oriented matroids, and polytopes. Any count-able category is an induced subcategory of separoids when they are endowed with homomorphisms (viz., mappingsthat preserve the so-called minimal Radon partitions).In this general framework, some results and invariants of different categories turn out to be special cases of thesame aspect; e.g., the pseudoachromatic number from graph theory and the Tverberg theorem from combinatorialconvexity are simply two faces of the same aspect, namely, complete colouring of separoids.

11.1 The axioms

A separoid is a set S endowed with a binary relation | ⊆ 2S × 2S on its power set, which satisfies the followingsimple properties for A,B ⊆ S :

A | B ⇔ B | A,

A | B ⇒ A ∩B = ∅,

A | B and A′ ⊂ A ⇒ A′ | B.

A related pair A | B is called a separation and we often say that A is separated from B. It is enough to know themaximal separations to reconstruct the separoid.A mapping φ : S → T is a morphism of separoids if the preimages of separations are separations; that is, forA,B ⊆ T

A | B ⇒ φ−1(A) | φ−1(B).

11.2 Examples

Examples of separoids can be found in almost every branch of mathematics. Here we list just a few.1. Given a graph G=(V,E), we can define a separoid on its vertices by saying that two (disjoint) subsets of V, say Aand B, are separated if there are no edges going from one to the other; i.e.,

A | B ⇔ ∀a ∈ A and b ∈ B : ab ̸∈ E.

28

11.3. THE BASIC LEMMA 29

2. Given an oriented matroid M = (E,T), given in terms of its topes T, we can define a separoid on E by saying thattwo subsets are separated if they are contained in opposite signs of a tope. In other words, the topes of an orientedmatroid are the maximal separations of a separoid. This example includes, of course, all directed graphs.3. Given a family of objects in an Euclidean space, we can define a separoid in it by saying that two subsets areseparated if there exists a hyperplane that separates them; i.e., leaving them in the two opposite sides of it.4. Given a topological space, we can define a separoid saying that two subsets are separated if there exist two disjointopen sets which contains them (one for each of them).

11.3 The basic lemma

Every separoid can be represented with a family of convex sets in some Euclidean space and their separations byhyperplanes.

11.4 References• Strausz Ricardo; “Separoides”. Situs, serie B, no. 5 (1998), Universidad Nacional Autónoma de México.

• Arocha Jorge Luis, Bracho Javier, Montejano Luis, Oliveros Deborah, Strausz Ricardo; “Separoids, their cat-egories and a Hadwiger-type theorem for transversals”. Discrete and Computational Geometry 27 (2002), no.3, 377–385.

• Strausz Ricardo; “Separoids and a Tverberg-type problem”. Geombinatorics 15 (2005), no. 2, 79–92.

• Montellano-Ballesteros Juan Jose, Por Attila, Strausz Ricardo; “Tverberg-type theorems for separoids”. Dis-crete and Computational Geometry 35 (2006), no.3, 513–523.

• Nešetřil Jaroslav, Strausz Ricardo; “Universality of separoids”. ArchivumMathematicum (Brno) 42 (2006), no.1, 85–101.

• Bracho Javier, Strausz Ricardo; “Two geometric representations of separoids”. Periodica Mathematica Hun-garica 53 (2006), no. 1-2, 115–120.

• Strausz Ricardo; “Homomorphisms of separoids”. 6th Czech-Slovak International Symposium on Combina-torics, Graph Theory, Algorithms and Applications, 461–468, Electronic Notes on Discrete Mathematics 28,Elsevier, Amsterdam, 2007.

• Strausz Ricardo; “Edrös-Szekeres 'happy end'-type theorems for separoids”. European Journal of Combina-torics 29 (2008), no. 4, 1076–1085.

Chapter 12

Series-parallel partial order

Series composition

Parallel composition

A series-parallel partial order, shown as a Hasse diagram.

In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-

30

12.1. DEFINITION 31

parallel partial orders by two simple composition operations.[1][2]

The series-parallel partial orders may be characterized as the N-free finite partial orders; they have order dimension atmost two.[1][3] They include weak orders and the reachability relationship in directed trees and directed series-parallelgraphs.[2][3] The comparability graphs of series-parallel partial orders are cographs.[2][4]

Series-parallel partial orders have been applied in job shop scheduling,[5] machine learning of event sequencing intime series data,[6] transmission sequencing of multimedia data,[7] and throughput maximization in dataflow pro-gramming.[8]

Series-parallel partial orders have also been called multitrees;[4] however, that name is ambiguous: multitrees alsorefer to partial orders with no four-element diamond suborder[9] and to other structures formed from multiple trees.

12.1 Definition

Consider P and Q, two partially ordered sets. The series composition of P and Q, written P; Q,[7] P * Q,[2] or P ⧀Q,[1]is the partially ordered set whose elements are the disjoint union of the elements of P andQ. In P;Q, two elementsx and y that both belong to P or that both belong to Q have the same order relation that they do in P or Q respectively.However, for every pair x, y where x belongs to P and y belongs to Q, there is an additional order relation x ≤ y in theseries composition. Series composition is an associative operation: one can write P; Q; R as the series compositionof three orders, without ambiguity about how to combine them pairwise, because both of the parenthesizations (P;Q); R and P; (Q; R) describe the same partial order. However, it is not a commutative operation, because switchingthe roles of P and Q will produce a different partial order that reverses the order relations of pairs with one elementin P and one in Q.[1]

The parallel composition of P and Q, written P || Q,[7] P + Q,[2] or P ⊕ Q,[1] is defined similarly, from the disjointunion of the elements in P and the elements in Q, with pairs of elements that both belong to P or both to Q havingthe same order as they do in P or Q respectively. In P || Q, a pair x, y is incomparable whenever x belongs to P and ybelongs to Q. Parallel composition is both commutative and associative.[1]

The class of series-parallel partial orders is the set of partial orders that can be built up from single-element partialorders using these two operations. Equivalently, it is the smallest set of partial orders that includes the single-elementpartial order and is closed under the series and parallel composition operations.[1][2]

A weak order is the series parallel partial order obtained from a sequence of composition operations in which allof the parallel compositions are performed first, and then the results of these compositions are combined using onlyseries compositions.[2]

12.2 Forbidden suborder characterization

The partial order N with the four elements a, b, c, and d and exactly the three order relations a ≤ b ≥ c ≤ d is anexample of a fence or zigzag poset; its Hasse diagram has the shape of the capital letter “N”. It is not series-parallel,because there is no way of splitting it into the series or parallel composition of two smaller partial orders. A partialorder P is said to be N-free if there does not exist a set of four elements in P such that the restriction of P to thoseelements is order-isomorphic to N. The series-parallel partial orders are exactly the nonempty finite N-free partialorders.[1][2][3]

It follows immediately from this (although it can also be proven directly) that any nonempty restriction of a series-parallel partial order is itself a series-parallel partial order.[1]

12.3 Order dimension

The order dimension of a partial order P is the minimum size of a realizer of P, a set of linear extensions of P withthe property that, for every two distinct elements x and y of P, x ≤ y in P if and only if x has an earlier position thany in every linear extension of the realizer. Series-parallel partial orders have order dimension at most two. If P andQ have realizers {L1, L2} and {L3, L4}, respectively, then {L1L3, L2L4} is a realizer of the series composition P; Q,and {L1L3, L4L2} is a realizer of the parallel composition P || Q.[2][3] A partial order is series-parallel if and only ifit has a realizer in which one of the two permutations is the identity and the other is a separable permutation.

32 CHAPTER 12. SERIES-PARALLEL PARTIAL ORDER

It is known that a partial order P has order dimension two if and only if there exists a conjugate order Q on thesame elements, with the property that any two distinct elements x and y are comparable on exactly one of these twoorders. In the case of series parallel partial orders, a conjugate order that is itself series parallel may be obtained byperforming a sequence of composition operations in the same order as the ones defining P on the same elements,but performing a series composition for each parallel composition in the decomposition of P and vice versa. Morestrongly, although a partial order may have many different conjugates, every conjugate of a series parallel partial ordermust itself be series parallel.[2]

12.4 Connections to graph theory

Any partial order may be represented (usually in more than one way) by a directed acyclic graph in which there is apath from x to ywhenever x and y are elements of the partial order with x ≤ y. The graphs that represent series-parallelpartial orders in this way have been called vertex series parallel graphs, and their transitive reductions (the graphsof the covering relations of the partial order) are called minimal vertex series parallel graphs.[3] Directed trees and(two-terminal) series parallel graphs are examples of minimal vertex series parallel graphs; therefore, series parallelpartial orders may be used to represent reachability relations in directed trees and series parallel graphs.[2][3]

The comparability graph of a partial order is the undirected graph with a vertex for each element and an undirectededge for each pair of distinct elements x, y with either x ≤ y or y ≤ x. That is, it is formed from a minimal vertexseries parallel graph by forgetting the orientation of each edge. The comparability graph of a series-partial order is acograph: the series and parallel composition operations of the partial order give rise to operations on the comparabilitygraph that form the disjoint union of two subgraphs or that connect two subgraphs by all possible edges; these twooperations are the basic operations from which cographs are defined. Conversely, every cograph is the comparabilitygraph of a series-parallel partial order. If a partial order has a cograph as its comparability graph, then it must be aseries-parallel partial order, because every other kind of partial order has an N suborder that would correspond to aninduced four-vertex path in its comparability graph, and such paths are forbidden in cographs.[2][4]

12.5 Computational complexity

It is possible to use the forbidden suborder characterization of series-parallel partial orders as a basis for an algorithmthat tests whether a given binary relation is a series-parallel partial order, in an amount of time that is linear inthe number of related pairs.[2][3] Alternatively, if a partial order is described as the reachability order of a directedacyclic graph, it is possible to test whether it is a series-parallel partial order, and if so compute its transitive closure,in time proportional to the number of vertices and edges in the transitive closure; it remains open whether the timeto recognize series-parallel reachability orders can be improved to be linear in the size of the input graph.[10]

If a series-parallel partial order is represented as an expression tree describing the series and parallel compositionoperations that formed it, then the elements of the partial order may be represented by the leaves of the expressiontree. A comparison between any two elements may be performed algorithmically by searching for the lowest commonancestor of the corresponding two leaves; if that ancestor is a parallel composition, the two elements are incomparable,and otherwise the order of the series composition operands determines the order of the elements. In this way, a series-parallel partial order on n elements may be represented in O(n) space with O(1) time to determine any comparisonvalue.[2]

It is NP-complete to test, for two given series-parallel partial orders P and Q, whether P contains a restriction iso-morphic to Q.[3]

Although the problem of counting the number of linear extensions of an arbitrary partial order is #P-complete,[11] itmay be solved in polynomial time for series-parallel partial orders. Specifically, if L(P) denotes the number of linearextensions of a partial order P, then L(P; Q) = L(P)L(Q) and

L(P ||Q) =(|P |+ |Q|)!|P |!|Q|!

L(P )L(Q),

so the number of linear extensions may be calculated using an expression tree with the same form as the decompositiontree of the given series-parallel order.[2]

12.6. APPLICATIONS 33

12.6 Applications

Mannila & Meek (2000) use series-parallel partial orders as a model for the sequences of events in time series data.They describe machine learning algorithms for inferring models of this type, and demonstrate its effectiveness atinferring course prerequisites from student enrollment data and at modeling web browser usage patterns.[6]

Amer et al. (1994) argue that series-parallel partial orders are a good fit for modeling the transmission sequencingrequirements of multimedia presentations. They use the formula for computing the number of linear extensions of aseries-parallel partial order as the basis for analyzing multimedia transmission algorithms.[7]

Choudhary et al. (1994) use series-parallel partial orders to model the task dependencies in a dataflow model ofmassive data processing for computer vision. They show that, by using series-parallel orders for this problem, it ispossible to efficiently construct an optimized schedule that assigns different tasks to different processors of a parallelcomputing system in order to optimize the throughput of the system.[8]

A class of orderings somewhat more general than series-parallel partial orders is provided by PQ trees, data structuresthat have been applied in algorithms for testing whether a graph is planar and recognizing interval graphs.[12] A Pnode of a PQ tree allows all possible orderings of its children, like a parallel composition of partial orders, while a Qnode requires the children to occur in a fixed linear ordering, like a series composition of partial orders. However,unlike series-parallel partial orders, PQ trees allow the linear ordering of any Q node to be reversed.

12.7 See also• Series and parallel circuits

12.8 References[1] Bechet, Denis; De Groote, Philippe; Retoré, Christian (1997), “A complete axiomatisation for the inclusion of series-

parallel partial orders”, Rewriting Techniques and Applications, Lecture Notes in Computer Science 1232, Springer-Verlag,pp. 230–240, doi:10.1007/3-540-62950-5_74.

[2] Möhring, Rolf H. (1989), “Computationally tractable classes of ordered sets”, in Rival, Ivan, Algorithms and Order: Pro-ceedings of the NATO Advanced Study Institute on Algorithms and Order, Ottawa, Canada, May 31-June 13, 1987, NATOScience Series C 255, Springer-Verlag, pp. 105–194, ISBN 978-0-7923-0007-6.

[3] Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982), “The recognition of series parallel digraphs”, SIAM Journalon Computing 11 (2): 298–313, doi:10.1137/0211023.

[4] Jung, H. A. (1978), “On a class of posets and the corresponding comparability graphs”, Journal of Combinatorial Theory,Series B 24 (2): 125–133, doi:10.1016/0095-8956(78)90013-8, MR 0491356.

[5] Lawler, Eugene L. (1978), “Sequencing jobs to minimize total weighted completion time subject to precedence constraints”,Annals of Discrete Mathematics 2: 75–90, doi:10.1016/S0167-5060(08)70323-6, MR 0495156.

[6] Mannila, Heikki; Meek, Christopher (2000), “Global partial orders from sequential data”, Proc. 6th ACM SIGKDD Inter-national Conference on Knowledge Discovery and Data Mining (KDD 2000), pp. 161–168, doi:10.1145/347090.347122.

[7] Amer, Paul D.; Chassot, Christophe; Connolly, Thomas J.; Diaz, Michel; Conrad, Phillip (1994), “Partial-order transportservice for multimedia and other applications”, IEEE/ACMTransactions onNetworking 2 (5): 440–456, doi:10.1109/90.336326.

[8] Choudhary, A. N.; Narahari, B.; Nicol, D. M.; Simha, R. (1994), “Optimal processor assignment for a class of pipelinedcomputations”, IEEE Transactions on Parallel and Distributed Systems 5 (4): 439–445, doi:10.1109/71.273050.

[9] Furnas, George W.; Zacks, Jeff (1994), “Multitrees: enriching and reusing hierarchical structure”, Proc. SIGCHI conferenceon Human Factors in Computing Systems (CHI '94), pp. 330–336, doi:10.1145/191666.191778.

[10] Ma, Tze-Heng; Spinrad, Jeremy (1991), “Transitive closure for restricted classes of partial orders”, Order 8 (2): 175–183,doi:10.1007/BF00383402.

[11] Brightwell, Graham R.; Winkler, Peter (1991), “Counting linear extensions”,Order 8 (3): 225–242, doi:10.1007/BF00383444.

[12] Booth, Kellogg S.; Lueker, George S. (1976), “Testing for the consecutive ones property, interval graphs, and graphplanarity using PQ-tree algorithms”, Journal of Computer and System Sciences 13 (3): 335–379, doi:10.1016/S0022-0000(76)80045-1.

Chapter 13

Surjective function

“Onto” redirects here. For other uses, see wikt:onto.In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if every element y in Y

X1

2

3

4

YD

B

C

A surjective function from domain X to codomain Y. The function is surjective because every point in the codomain is the value off(x) for at least one point x in the domain.

34

13.1. DEFINITION 35

has a corresponding element x in X such that f(x) = y. The function f may map more than one element of X to thesame element of Y.The term surjective and the related terms injective and bijectivewere introduced by Nicolas Bourbaki,[1] the pseudonymfor a group of mainly French 20th-century mathematicians who wrote a series of books presenting an exposition ofmodern advanced mathematics, beginning in 1935. The French prefix sur means over or above and relates to the factthat the image of the domain of a surjective function completely covers the function’s codomain.

13.1 Definition

For more details on notation, see Function (mathematics) § Notation.

A surjective function is a function whose image is equal to its codomain. Equivalently, a function f with domainX and codomain Y is surjective if for every y in Y there exists at least one x in X with f(x) = y . Surjections aresometimes denoted by a two-headed rightwards arrow (U+21A0 ↠ rightwards two headed arrow),[2] as in f : X ↠Y.Symbolically,

If f : X → Y , then f is said to be surjective if

∀y ∈ Y, ∃x ∈ X, f(x) = y

13.2 Examples

For any set X, the identity function idX on X is surjective.The function f : Z → {0,1} defined by f(n) = n mod 2 (that is, even integers are mapped to 0 and odd integers to 1)is surjective.The function f : R → R defined by f(x) = 2x + 1 is surjective (and even bijective), because for every real number ywe have an x such that f(x) = y: an appropriate x is (y − 1)/2.The function f : R → R defined by f(x) = x3 − 3x is surjective, because the pre-image of any real number y is thesolution set of the cubic polynomial equation x3 − 3x − y = 0 and every cubic polynomial with real coefficients has atleast one real root. However, this function is not injective (and hence not bijective) since e.g. the pre-image of y = 2is {x = −1, x = 2}. (In fact, the pre-image of this function for every y, −2 ≤ y ≤ 2 has more than one element.)The function g : R → R defined by g(x) = x2 is not surjective, because there is no real number x such that x2 = −1.However, the function g : R → R0

+ defined by g(x) = x2 (with restricted codomain) is surjective because for every yin the nonnegative real codomain Y there is at least one x in the real domain X such that x2 = y.The natural logarithm function ln : (0,+∞) → R is a surjective and even bijective mapping from the set of positivereal numbers to the set of all real numbers. Its inverse, the exponential function, is not surjective as its range is the setof positive real numbers and its domain is usually defined to be the set of all real numbers. The matrix exponentialis not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as amap from the space of all n×n matrices to the general linear group of degree n, i.e. the group of all n×n invertiblematrices. Under this definition the matrix exponential is surjective for complex matrices, although still not surjectivefor real matrices.The projection from a cartesian product A × B to one of its factors is surjective unless the other factor is empty.In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function.

13.3 Properties

A function is bijective if and only if it is both surjective and injective.

36 CHAPTER 13. SURJECTIVE FUNCTION

X Y

f(x)

f : X → Y

x

A non-surjective function from domain X to codomain Y. The smaller oval inside Y is the image (also called range) of f. Thisfunction is not surjective, because the image does not fill the whole codomain. In other words, Y is colored in a two-step process:First, for every x in X, the point f(x) is colored yellow; Second, all the rest of the points in Y, that are not yellow, are colored blue.The function f is surjective only if there are no blue points.

If (as is often done) a function is identified with its graph, then surjectivity is not a property of the function itself, butrather a relationship between the function and its codomain. Unlike injectivity, surjectivity cannot be read off of thegraph of the function alone.

13.3.1 Surjections as right invertible functions

The function g : Y → X is said to be a right inverse of the function f : X → Y if f(g(y)) = y for every y in Y (g can beundone by f). In other words, g is a right inverse of f if the composition f o g of g and f in that order is the identityfunction on the domain Y of g. The function g need not be a complete inverse of f because the composition in theother order, g o f, may not be the identity function on the domain X of f. In other words, f can undo or "reverse" g,but cannot necessarily be reversed by it.Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has aright inverse is equivalent to the axiom of choice.If f : X → Y is surjective and B is a subset of Y, then f(f −1(B)) = B. Thus, B can be recovered from its preimagef −1(B).For example, in the first illustration, there is some function g such that g(C) = 4. There is also some function f suchthat f(4) = C. It doesn't matter that g(C) can also equal 3; it only matters that f “reverses” g.

• Another surjective function. (This one happens to be a bijection)

• A non-surjective function. (This one happens to be an injection)

• Surjective composition: the first function need not be surjective.

13.3. PROPERTIES 37

x

y

x

y

1 X 1 Y : f 2 X 2 Y : f

2 X x 1 X x

x f y

f im Y y

Y y

f im

Y X : f x f y

X x

Interpretation for surjective functions in the Cartesian plane, defined by the mapping f : X → Y, where y = f(x), X = domain offunction, Y = range of function. Every element in the range is mapped onto from an element in the domain, by the rule f. Theremay be a number of domain elements which map to the same range element. That is, every y in Y is mapped from an element x inX, more than one x can map to the same y. Left: Only one domain is shown which makes f surjective. Right: two possible domainsX1 and X2 are shown.

x

y

X x 0

Y X : f x f y

X x

Y y

f im

Y y 0

x

y

X x 1

Y y 2

Y y 1

X x 2

X x 3

Y y 3

X x

1 X 1 Y : f 2 X 2 Y : f

Y y

f im

2 X x 1 X x

x f y

Non-surjective functions in the Cartesian plane. Although some parts of the function are surjective, where elements y in Y do havea value x in X such that y = f(x), some parts are not. Left: There is y0 in Y, but there is no x0 in X such that y0 = f(x0). Right:There are y1, y2 and y3 in Y, but there are no x1, x2, and x3 in X such that y1 = f(x1), y2 = f(x2), and y3 = f(x3).

13.3.2 Surjections as epimorphisms

A function f : X → Y is surjective if and only if it is right-cancellative:[3] given any functions g,h : Y → Z, whenever go f = h o f, then g = h. This property is formulated in terms of functions and their composition and can be generalizedto the more general notion of the morphisms of a category and their composition. Right-cancellative morphisms arecalled epimorphisms. Specifically, surjective functions are precisely the epimorphisms in the category of sets. The

38 CHAPTER 13. SURJECTIVE FUNCTION

prefix epi is derived from the Greek preposition ἐπί meaning over, above, on.Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse g of amorphism f is called a section of f. A morphism with a right inverse is called a split epimorphism.

13.3.3 Surjections as binary relations

Any function with domain X and codomain Y can be seen as a left-total and right-unique binary relation between Xand Y by identifying it with its function graph. A surjective function with domain X and codomain Y is then a binaryrelation between X and Y that is right-unique and both left-total and right-total.

13.3.4 Cardinality of the domain of a surjection

The cardinality of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If f: X → Y is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers. (Theproof appeals to the axiom of choice to show that a function g : Y → X satisfying f(g(y)) = y for all y in Y exists. gis easily seen to be injective, thus the formal definition of |Y | ≤ |X| is satisfied.)Specifically, if both X and Y are finite with the same number of elements, then f : X → Y is surjective if and only iff is injective.

13.3.5 Composition and decomposition

The composite of surjective functions is always surjective: If f and g are both surjective, and the codomain of gis equal to the domain of f, then f o g is surjective. Conversely, if f o g is surjective, then f is surjective (but g,the function applied first, need not be). These properties generalize from surjections in the category of sets to anyepimorphisms in any category.Any function can be decomposed into a surjection and an injection: For any function h : X→ Z there exist a surjectionf : X → Y and an injection g : Y → Z such that h = g o f. To see this, define Y to be the sets h −1(z) where z is inZ. These sets are disjoint and partition X. Then f carries each x to the element of Y which contains it, and g carrieseach element of Y to the point in Z to which h sends its points. Then f is surjective since it is a projection map, andg is injective by definition.

13.3.6 Induced surjection and induced bijection

Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijectiondefined on a quotient of its domain by collapsing all arguments mapping to a given fixed image. More precisely, everysurjection f : A → B can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalenceclasses of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set ofall preimages under f. Let P(~) : A → A/~ be the projection map which sends each x in A to its equivalence class[x]~, and let fP : A/~ → B be the well-defined function given by fP([x]~) = f(x). Then f = fP o P(~).

13.4 See also• Bijection, injection and surjection

• Cover (algebra)

• Covering map

• Enumeration

• Fiber bundle

• Index set

• Section (category theory)

13.5. NOTES 39

13.5 Notes[1] “Injection, Surjection and Bijection”, Earliest Uses of Some of the Words of Mathematics, Tripod.

[2] “Arrows – Unicode” (PDF). Retrieved 2013-05-11.

[3] Goldblatt, Robert (2006) [1984]. Topoi, the Categorial Analysis of Logic (Revised ed.). Dover Publications. ISBN 978-0-486-45026-1. Retrieved 2009-11-25.

13.6 References• Bourbaki, Nicolas (2004) [1968]. Theory of Sets. Springer. ISBN 978-3-540-22525-6.

Chapter 14

Symmetric closure

In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation on X thatcontains R.For example, if X is a set of airports and xRy means “there is a direct flight from airport x to airport y", then thesymmetric closure of R is the relation “there is a direct flight either from x to y or from y to x". Or, if X is the set ofhumans (alive or dead) and R is the relation 'parent of', then the symmetric closure of R is the relation "x is a parentor a child of y".

14.1 Definition

The symmetric closure S of a relation R on a set X is given by

S = R ∪ {(x, y) : (y, x) ∈ R} .

In other words, the symmetric closure of R is the union of R with its inverse relation, R−1.

14.2 See also• Transitive closure

• Reflexive closure

14.3 References• Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8

40

Chapter 15

Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that if ais related to b then b is related to a.In mathematical notation, this is:

∀a, b ∈ X, aRb ⇒ bRa.

15.1 Examples

15.1.1 In mathematics

• “is equal to” (equality) (whereas “is less than” is not symmetric)

• “is comparable to”, for elements of a partially ordered set

• "... and ... are odd":

41

42 CHAPTER 15. SYMMETRIC RELATION

15.1.2 Outside mathematics

• “is married to” (in most legal systems)

• “is a fully biological sibling of”

• “is a homophone of”

15.2 Relationship to asymmetric and antisymmetric relations

By definition, a relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be relatedto a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for “is lessthan or equal to” and “preys on”).Symmetric and antisymmetric (where the only way a can be related to b and b be related to a is if a = b) are actuallyindependent of each other, as these examples show.

15.3 Additional aspects

A symmetric relation that is also transitive and reflexive is an equivalence relation.One way to conceptualize a symmetric relation in graph theory is that a symmetric relation is an edge, with the edge’stwo vertices being the two entities so related. Thus, symmetric relations and undirected graphs are combinatoriallyequivalent objects.

15.4. SEE ALSO 43

15.4 See also• Symmetry in mathematics

• Symmetry

• Asymmetric relation

• Antisymmetric relation

Chapter 16

Ternary equivalence relation

In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalencerelation. A ternary equivalence relation is symmetric, reflexive, and transitive. The classic example is the relationof collinearity among three points in Euclidean space. In an abstract set, a ternary equivalence relation determines acollection of equivalence classes or pencils that form a linear space in the sense of incidence geometry. In the sameway, a binary equivalence relation on a set determines a partition.

16.1 Definition

A ternary equivalence relation on a set X is a relation E ⊂ X3, written [a, b, c], that satisfies the following axioms:

1. Symmetry: If [a, b, c] then [b, c, a] and [c, b, a]. (Therefore also [a, c, b], [b, a, c], and [c, a, b].)

2. Reflexivity: [a, b, b]. Equivalently, if a, b, and c are not all distinct, then [a, b, c].

3. Transitivity: If a ≠ b and [a, b, c] and [a, b, d] then [b, c, d]. (Therefore also [a, c, d].)

16.2 References• Araújoa, João; Koniecznyc, Janusz (2007), “A method of finding automorphism groups of endomorphism

monoids of relational systems”, Discrete Mathematics 307: 1609–1620, doi:10.1016/j.disc.2006.09.029

• Bachmann, Friedrich, Aufbau der Geometrie aus dem Spiegelungsbegriff

• Karzel, Helmut (2007), “Loops related to geometric structures”, Quasigroups and Related Systems 15: 47−76

• Karzel, Helmut; Pianta, Silvia (2008), “Binary operations derived from symmetric permutation sets and appli-cations to absolute geometry”, Discrete Mathematics 308: 415–421, doi:10.1016/j.disc.2006.11.058

• Karzel, Helmut; Marchi, Mario; Pianta, Silvia (December 2010), “The defect in an invariant reflection struc-ture”, Journal of Geometry 99 (1-2): 67–87, doi:10.1007/s00022-010-0058-7

• Lingenberg, Rolf (1979), Metric planes and metric vector spaces, Wiley

• Rainich, G.Y. (1952), “Ternary relations in geometry and algebra”, Michigan Mathematical Journal 1 (2):97–111, doi:10.1307/mmj/1028988890

• Szmielew, Wanda (1981), On n-ary equivalence relations and their application to geometry, Warsaw: InstytutMatematyczny Polskiej Akademi Nauk

44

Chapter 17

Ternary relation

In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in therelation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place.Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian product A × B of some setsA and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product A × B × C of three sets A,B and C.An example of a ternary relation in elementary geometry is the collinearity of points.

17.1 Examples

17.1.1 Binary functions

Further information: Graph of a function and binary function

A function ƒ: A × B → C in two variables, taking values in two sets A and B, respectively, is formally a function thatassociates to every pair (a,b) in A × B an element ƒ(a, b) in C. Therefore its graph consists of pairs of the form ((a,b), ƒ(a, b)). Such pairs in which the first element is itself a pair are often identified with triples. This makes the graphof ƒ a ternary relation between A, B and C, consisting of all triples (a, b, ƒ(a, b)), for all a in A and b in B.

17.1.2 Cyclic orders

Main article: Cyclic order

Given any set A whose elements are arranged on a circle, one can define a ternary relation R on A, i.e. a subset of A3

= A × A × A, by stipulating that R(a, b, c) holds if and only if the elements a, b and c are pairwise different and whengoing from a to c in a clockwise direction one passes through b. For example if A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12 } represents the hours on a clock face, then R(8, 12, 4) holds and R(12, 8, 4) does not hold.

17.1.3 Betweenness relations

Main article: Betweenness relation

17.1.4 Congruence relation

Main article: Congruence modulo m

45

46 CHAPTER 17. TERNARY RELATION

The ordinary congruence of arithmetics

a ≡ b (mod m)

which holds for three integers a, b, and m if and only if m divides a − b, formally may be considered as a ternaryrelation. However, usually, this instead is considered as a family of binary relations between the a and the b, indexedby the modulusm. For each fixedm, indeed this binary relation has some natural properties, like being an equivalencerelation; while the combined ternary relation in general is not studied as one relation.

17.1.5 Typing relation

Main article: Simply typed lambda calculus § Typing rules

A typing relation Γ ⊢ e : σ indicates that e is a term of type σ in context Γ , and is thus a ternary relation betweencontexts, terms and types.

17.2 Further reading• Myers, Dale (1997), “An interpretive isomorphism between binary and ternary relations”, in Mycielski, Jan;

Rozenberg, Grzegorz; Salomaa, Arto, Structures in Logic and Computer Science, Lecture Notes in ComputerScience 1261, Springer, pp. 84–105, doi:10.1007/3-540-63246-8_6, ISBN 3-540-63246-8

• Novák, Vítězslav (1996), “Ternary structures and partial semigroups”, Czechoslovak Mathematical Journal 46(1): 111–120, hdl:10338.dmlcz/127275

• Novák, Vítězslav; Novotný, Miroslav (1989), “Transitive ternary relations and quasiorderings”, ArchivumMathematicum 25 (1–2): 5–12, hdl:10338.dmlcz/107333

• Novák, Vítězslav; Novotný, Miroslav (1992), “Binary and ternary relations”, Mathematica Bohemica 117 (3):283–292, hdl:10338.dmlcz/126278

• Novotný, Miroslav (1991), “Ternary structures and groupoids”, Czechoslovak Mathematical Journal 41 (1):90–98, hdl:10338.dmlcz/102437

• Šlapal, Josef (1993), “Relations and topologies”,CzechoslovakMathematical Journal 43 (1): 141–150, hdl:10338.dmlcz/128381

Chapter 18

Tolerance relation

In mathematics, a tolerance relation is a relation that is reflexive and symmetric. It does not need to be transitive.

18.1 External links• Gerasin, S. N., Shlyakhov, V. V., and Yakovlev, S. V. 2008. Set coverings and tolerance relations. Cybernetics

and Sys. Anal. 44, 3 (May 2008), 333–340. doi:10.1007/s10559-008-9007-y

• Hryniewiecki, K. 1991, Relations of Tolerance

FORMALIZED MATHEMATICS, Vol. 2, No. 1, January–February 1991.

47

Chapter 19

Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denotedby infix ≤) on some setX which is transitive, antisymmetric, and total. A set paired with a total order is called a totallyordered set, a linearly ordered set, a simply ordered set, or a chain.If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:

If a ≤ b and b ≤ a then a = b (antisymmetry);If a ≤ b and b ≤ c then a ≤ c (transitivity);a ≤ b or b ≤ a (totality).

Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a.[1] A relation having the property of“totality” means that any pair of elements in the set of the relation are comparable under the relation. This also meansthat the set can be diagrammed as a line of elements, giving it the name linear.[2] Totality also implies reflexivity, i.e.,a ≤ a. Therefore, a total order is also a partial order. The partial order has a weaker form of the third condition. (Itrequires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extensionof that partial order.

19.1 Strict total order

For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) relation <, called a strict totalorder, which can equivalently be defined in two ways:

• a < b if and only if a ≤ b and a ≠ b

• a < b if and only if not b ≤ a (i.e., < is the inverse of the complement of ≤)

Properties:

• The relation is transitive: a < b and b < c implies a < c.• The relation is trichotomous: exactly one of a < b, b < a and a = b is true.• The relation is a strict weak order, where the associated equivalence is equality.

We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤can equivalently be defined in two ways:

• a ≤ b if and only if a < b or a = b

• a ≤ b if and only if not b < a

Two more associated orders are the complements ≥ and >, completing the quadruple {<, >, ≤, ≥}.We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whetherwe are talking about the non-strict or the strict total order.

48

19.2. EXAMPLES 49

19.2 Examples

• The letters of the alphabet ordered by the standard dictionary order, e.g., A < B < C etc.

• Any subset of a totally ordered set, with the restriction of the order on the whole set.

• Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders).

• If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on Xby setting x1 < x2 if and only if f(x1) < f(x2).

• The lexicographical order on the Cartesian product of a set of totally ordered sets indexed by an ordinal, isitself a total order. For example, any set of words ordered alphabetically is a totally ordered set, viewed as asubset of a Cartesian product of a countable number of copies of a set formed by adding the space symbol tothe alphabet (and defining a space to be less than any letter).

• The set of real numbers ordered by the usual less than (<) or greater than (>) relations is totally ordered, hencealso the subsets of natural numbers, integers, and rational numbers. Each of these can be shown to be the unique(to within isomorphism) smallest example of a totally ordered set with a certain property, (a total order A isthe smallest with a certain property if whenever B has the property, there is an order isomorphism from A to asubset of B):

• The natural numbers comprise the smallest totally ordered set with no upper bound.• The integers comprise the smallest totally ordered set with neither an upper nor a lower bound.• The rational numbers comprise the smallest totally ordered set which is dense in the real numbers. The

definition of density used here says that for every 'a' and 'b' in the real numbers such that 'a' < 'b', there isa 'q' in the rational numbers such that 'a' < 'q' < 'b'.

• The real numbers comprise the smallest unbounded totally ordered set that is connected in the ordertopology (defined below).

• Ordered fields are totally ordered by definition. They include the rational numbers and the real numbers.

19.3 Further concepts

19.3.1 Chains

While chain is sometimes merely a synonym for totally ordered set, it can also refer to a totally ordered subset ofsome partially ordered set. The latter definition has a crucial role in Zorn’s lemma.For example, consider the set of all subsets of the integers partially ordered by inclusion. Then the set { In : n is anatural number}, where In is the set of natural numbers below n, is a chain in this ordering, as it is totally orderedunder inclusion: If n≤k, then In is a subset of Ik.

19.3.2 Lattice theory

One may define a totally ordered set as a particular kind of lattice, namely one in which we have

{a ∨ b, a ∧ b} = {a, b} for all a, b.

We then write a ≤ b if and only if a = a ∧ b . Hence a totally ordered set is a distributive lattice.

50 CHAPTER 19. TOTAL ORDER

19.3.3 Finite total orders

A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subsetthereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observingthat every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphicto an initial segment of the natural numbers ordered by <. In other words a total order on a set with k elements inducesa bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with ordertype ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).

19.3.4 Category theory

Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being mapswhich respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b).A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.

19.3.5 Order topology

For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b},(a, ∞) = {x : a < x} and (−∞, ∞) = X. We can use these open intervals to define a topology on any ordered set, theorder topology.When more than one order is being used on a set one talks about the order topology induced by a particular order.For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology onN induced by < and the order topology on N induced by > (in this case they happen to be identical but will not ingeneral).The order topology induced by a total order may be shown to be hereditarily normal.

19.3.6 Completeness

A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upperbound. For example, the set of real numbers R is complete but the set of rational numbers Q is not.There are a number of results relating properties of the order topology to the completeness of X:

• If the order topology on X is connected, X is complete.

• X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is twopoints a and b in X with a < b such that no c satisfies a < c < b.)

• X is complete if and only if every bounded set that is closed in the order topology is compact.

A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervalsof real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line).There are order-preserving homeomorphisms between these examples.

19.3.7 Sums of orders

For any two disjoint total orders (A1,≤1) and (A2,≤2) , there is a natural order ≤+ on the set A1 ∪ A2 , which iscalled the sum of the two orders or sometimes just A1 +A2 :

For x, y ∈ A1 ∪A2 , x ≤+ y holds if and only if one of the following holds:

1. x, y ∈ A1 and x ≤1 y

2. x, y ∈ A2 and x ≤2 y

3. x ∈ A1 and y ∈ A2

19.4. ORDERS ON THE CARTESIAN PRODUCT OF TOTALLY ORDERED SETS 51

Intutitively, this means that the elements of the second set are added on top of the elements of the first set.More generally, if (I,≤) is a totally ordered index set, and for each i ∈ I the structure (Ai,≤i) is a linear order,where the sets Ai are pairwise disjoint, then the natural total order on

∪iAi is defined by

For x, y ∈∪i∈I Ai , x ≤ y holds if:

1. Either there is some i ∈ I with x ≤i y2. or there are some i < j in I with x ∈ Ai , y ∈ Aj

19.4 Orders on the Cartesian product of totally ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product oftwo totally ordered sets are:

• Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order.• (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order.• (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of

the corresponding strict total orders). This is also a partial order.

All three can similarly be defined for the Cartesian product of more than two sets.Applied to the vector space Rn, each of these make it an ordered vector space.See also examples of partially ordered sets.A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding totalpreorder on that subset.

19.5 Related structures

A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order.A group with a compatible total order is a totally ordered group.There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientationresults in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both dataresults in a separation relation.[3]

19.6 See also• Order theory• Well-order• Suslin’s problem• Countryman line

19.7 Notes[1] Nederpelt, Rob (2004). “Chapter 20.2: Ordered Sets. Orderings”. Logical Reasoning: A First Course. Texts in Computing

3 (3rd, Revised ed.). King’s College Publications. p. 325. ISBN 0-9543006-7-X.

[2] Nederpelt, Rob (2004). “Chapter 20.3: Ordered Sets. Linear orderings”. Logical Reasoning: A First Course. Texts inComputing 3 (3rd, Revisied ed.). King’s College Publications. p. 330. ISBN 0-9543006-7-X.

[3] Macpherson, H. Dugald (2011), “A survey of homogeneous structures” (PDF),DiscreteMathematics, doi:10.1016/j.disc.2011.01.024,retrieved 28 April 2011

52 CHAPTER 19. TOTAL ORDER

19.8 References• George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN

0-7167-0442-0

• John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4

Chapter 20

Total relation

In mathematics, a binary relation R over a set X is total or complete if for all a and b in X, a is related to b or b isrelated to a (or both).In mathematical notation, this is

∀a, b ∈ X, aRb ∨ bRa.

Total relations are sometimes said to have comparability.

20.1 Examples

For example, “is less than or equal to” is a total relation over the set of real numbers, because for two numbers eitherthe first is less than or equal to the second, or the second is less than or equal to the first. On the other hand, “is lessthan” is not a total relation, since one can pick two equal numbers, and then neither the first is less than the second, noris the second less than the first. (But note that “is less than” is a weak order which gives rise to a total order, namely“is less than or equal to”. The relationship between strict orders and weak orders is discussed at partially ordered set.)The relation “is a subset of” is also not total because, for example, neither of the sets {1,2} and {3,4} is a subset ofthe other.

20.2 Properties and related notions

Totality implies reflexivity.If a transitive relation is also total, it is a total preorder. If a partial order is also total, it is a total order.A binary relation R over X is called connex if for all a and b in X such that a ≠ b, a is related to b or b is related to a(or both):[1]

∀a, b ∈ X, aRb ∨ bRa ∨ (a = b).

Connexity does not imply reflexivity. A strict partial order is a strict total order if and only if it is connex.

20.3 See also

• Total order

53

54 CHAPTER 20. TOTAL RELATION

20.4 References[1] Rautenberg, Wolfgang (2010),AConcise Introduction toMathematical Logic (3rd ed.), New York: Springer Science+Business

Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6

Chapter 21

Transitive closure

For other uses, see Closure (disambiguation).This article is about the transitive closure of a binary relation. For the transitive closure of a set, see transitiveset#Transitive closure.

In mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R+ on set X such thatR+ contains R and R+ is minimal (Lidl and Pilz 1998:337). If the binary relation itself is transitive, then the transitiveclosure is that same binary relation; otherwise, the transitive closure is a different relation. For example, if X is a setof airports and x R y means “there is a direct flight from airport x to airport y", then the transitive closure of R on Xis the relation R+: “it is possible to fly from x to y in one or more flights.”

21.1 Transitive relations and examples

A relation R on a set X is transitive if, for all x, y, z in X, whenever x R y and y R z then x R z. Examples of transitiverelations include the equality relation on any set, the “less than or equal” relation on any linearly ordered set, and therelation "x was born before y" on the set of all people. Symbolically, this can be denoted as: if x < y and y < z then x< z.One example of a non-transitive relation is “city x can be reached via a direct flight from city y" on the set of all cities.Simply because there is a direct flight from one city to a second city, and a direct flight from the second city to thethird, does not imply there is a direct flight from the first city to the third. The transitive closure of this relation is adifferent relation, namely “there is a sequence of direct flights that begins at city x and ends at city y". Every relationcan be extended in a similar way to a transitive relation.An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y".The transitive closure of this relation is “some day x comes after a day y on the calendar”, which is trivially true for alldays of the week x and y (and thus equivalent to the Cartesian square, which is "x and y are both days of the week”).

21.2 Existence and description

For any relation R, the transitive closure of R always exists. To see this, note that the intersection of any family oftransitive relations is again transitive. Furthermore, there exists at least one transitive relation containing R, namelythe trivial one: X × X. The transitive closure of R is then given by the intersection of all transitive relations containingR.For finite sets, we can construct the transitive closure step by step, starting from R and adding transitive edges. Thisgives the intuition for a general construction. For any set X, we can prove that transitive closure is given by thefollowing expression

R+ =∪

i∈{1,2,3,...}

Ri.

55

56 CHAPTER 21. TRANSITIVE CLOSURE

where Ri is the i-th power of R, defined inductively by

R1 = R

and, for i > 0 ,

Ri+1 = R ◦Ri

where ◦ denotes composition of relations.To show that the above definition of R+ is the least transitive relation containing R, we show that it contains R, that itis transitive, and that it is the smallest set with both of those characteristics.

• R ⊆ R+ : R+ contains all of the Ri , so in particular R+ contains R .

• R+ is transitive: every element of R+ is in one of the Ri , so R+ must be transitive by the following reasoning:if (s1, s2) ∈ Rj and (s2, s3) ∈ Rk , then from composition’s associativity, (s1, s3) ∈ Rj+k (and thus in R+

) because of the definition of Ri .

• R+ is minimal: Let G be any transitive relation containing R , we want to show that R+ ⊆ G . It is sufficient toshow that for every i > 0 , Ri ⊆ G . Well, since G contains R , R1 ⊆ G . And since G is transitive, wheneverRi ⊆ G , Ri+1 ⊆ G according to the construction of Ri and what it means to be transitive. Therefore, byinduction, G contains every Ri , and thus also R+ .

21.3 Properties

The intersection of two transitive relations is transitive.The union of two transitive relations need not be transitive. To preserve transitivity, one must take the transitiveclosure. This occurs, for example, when taking the union of two equivalence relations or two preorders. To obtain anew equivalence relation or preorder one must take the transitive closure (reflexivity and symmetry—in the case ofequivalence relations—are automatic).

21.4 In graph theory

In computer science, the concept of transitive closure can be thought of as constructing a data structure that makesit possible to answer reachability questions. That is, can one get from node a to node d in one or more hops? Abinary relation tells you only that node a is connected to node b, and that node b is connected to node c, etc. Afterthe transitive closure is constructed, as depicted in the following figure, in an O(1) operation one may determine thatnode d is reachable from node a. The data structure is typically stored as a matrix, so if matrix[1][4] = 1, then it isthe case that node 1 can reach node 4 through one or more hops.The transitive closure of a directed acyclic graph (DAG) is the reachability relation of the DAG and a strict partialorder.

21.5 In logic and computational complexity

The transitive closure of a binary relation cannot, in general, be expressed in first-order logic (FO). This means thatone cannot write a formula using predicate symbols R and T that will be satisfied in any model if and only if T isthe transitive closure of R. In finite model theory, first-order logic (FO) extended with a transitive closure operatoris usually called transitive closure logic, and abbreviated FO(TC) or just TC. TC is a sub-type of fixpoint logics.The fact that FO(TC) is strictly more expressive than FO was discovered by Ronald Fagin in 1974; the result wasthen rediscovered by Alfred Aho and Jeffrey Ullman in 1979, who proposed to use fixpoint logic as a database query

21.6. IN DATABASE QUERY LANGUAGES 57

Output

Input

Transitive closure constructs the output graph from the input graph.

language (Libkin 2004:vii). With more recent concepts of finite model theory, proof that FO(TC) is strictly moreexpressive than FO follows immediately from the fact that FO(TC) is not Gaifman-local (Libkin 2004:49).In computational complexity theory, the complexity class NL corresponds precisely to the set of logical sentencesexpressible in TC. This is because the transitive closure property has a close relationship with the NL-completeproblem STCON for finding directed paths in a graph. Similarly, the class L is first-order logic with the commutative,transitive closure. When transitive closure is added to second-order logic instead, we obtain PSPACE.

21.6 In database query languages

Further information: Hierarchical and recursive queries in SQL

Since the 1980s Oracle Database has implemented a proprietary SQL extension CONNECT BY... START WITHthat allows the computation of a transitive closure as part of a declarative query. The SQL 3 (1999) standard addeda more general WITH RECURSIVE construct also allowing transitive closures to be computed inside the queryprocessor; as of 2011 the latter is implemented in IBM DB2, Microsoft SQL Server, and PostgreSQL, although notin MySQL (Benedikt and Senellart 2011:189).Datalog also implements transitive closure computations (Silberschatz et al. 2010:C.3.6).

21.7 Algorithms

Efficient algorithms for computing the transitive closure of a graph can be found in Nuutila (1995). The fastest worst-case methods, which are not practical, reduce the problem to matrix multiplication. The problem can also be solvedby the Floyd–Warshall algorithm, or by repeated breadth-first search or depth-first search starting from each node ofthe graph.More recent research has explored efficient ways of computing transitive closure on distributed systems based on theMapReduce paradigm (Afrati et al. 2011).

21.8 See also• Deductive closure

58 CHAPTER 21. TRANSITIVE CLOSURE

• Transitive reduction (a smallest relation having the transitive closure of R as its transitive closure)

• Symmetric closure

• Reflexive closure

• Ancestral relation

21.9 References• Lidl, R. and Pilz, G., 1998, Applied abstract algebra, 2nd edition, Undergraduate Texts in Mathematics,

Springer, ISBN 0-387-98290-6

• Keller, U., 2004, Some Remarks on the Definability of Transitive Closure in First-order Logic and Datalog(unpublished manuscript)

• Erich Grädel; Phokion G. Kolaitis; Leonid Libkin; Maarten Marx; Joel Spencer; Moshe Y. Vardi; Yde Venema;Scott Weinstein (2007). Finite Model Theory and Its Applications. Springer. pp. 151–152. ISBN 978-3-540-68804-4.

• Libkin, Leonid (2004), Elements of Finite Model Theory, Springer, ISBN 978-3-540-21202-7

• Heinz-Dieter Ebbinghaus; Jörg Flum (1999). FiniteModel Theory (2nd ed.). Springer. pp. 123–124, 151–161,220–235. ISBN 978-3-540-28787-2.

• Aho, A. V.; Ullman, J. D. (1979). “Universality of data retrieval languages”. Proceedings of the 6th ACMSIGACT-SIGPLAN Symposium on Principles of programming languages - POPL '79. p. 110. doi:10.1145/567752.567763.

• Benedikt, M.; Senellart, P. (2011). “Databases”. In Blum, Edward K.; Aho, Alfred V. Computer Science. TheHardware, Software and Heart of It. pp. 169–229. doi:10.1007/978-1-4614-1168-0_10. ISBN 978-1-4614-1167-3.

• Nuutila, E., Efficient Transitive Closure Computation in Large Digraphs. Acta Polytechnica Scandinavica,Mathematics and Computing in Engineering Series No. 74, Helsinki 1995, 124 pages. Published by theFinnish Academy of Technology. ISBN 951-666-451-2, ISSN 1237-2404, UDC 681.3.

• Abraham Silberschatz; Henry Korth; S. Sudarshan (2010). Database System Concepts (6th ed.). McGraw-Hill.ISBN 978-0-07-352332-3. Appendix C (online only)

• Foto N. Afrati, Vinayak Borkar, Michael Carey, Neoklis Polyzotis, Jeffrey D. Ullman, Map-Reduce Extensionsand Recursive Queries, EDBT 2011, March 22–24, 2011, Uppsala, Sweden, ISBN 978-1-4503-0528-0

21.10 External links• "Transitive closure and reduction", The Stony Brook Algorithm Repository, Steven Skiena .

• "Apti Algoritmi", An example and some C++ implementations of algorithms that calculate the transitive closureof a given binary relation, Vreda Pieterse.

Chapter 22

Transitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b,and b is in turn related to an element c, then a is also related to c. Transitivity is a key property of both partial orderrelations and equivalence relations.

22.1 Formal definition

In terms of set theory, the transitive relation can be defined as:

∀a, b, c ∈ X : (aRb ∧ bRc) ⇒ aRc

22.2 Examples

For example, “is greater than,” “is at least as great as,” and “is equal to” (equality) are transitive relations:

whenever A > B and B > C, then also A > Cwhenever A ≥ B and B ≥ C, then also A ≥ Cwhenever A = B and B = C, then also A = C.

On the other hand, “is the mother of” is not a transitive relation, because if Alice is the mother of Brenda, and Brendais the mother of Claire, then Alice is not the mother of Claire. What is more, it is antitransitive: Alice can never bethe mother of Claire.Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relation“is a matrilinear ancestor of”. This is a transitive relation. More precisely, it is the transitive closure of the relation“is the mother of”.More examples of transitive relations:

• “is a subset of” (set inclusion)

• “divides” (divisibility)

• “implies” (implication)

22.3 Properties

59

60 CHAPTER 22. TRANSITIVE RELATION

22.3.1 Closure properties

The converse of a transitive relation is always transitive: e.g. knowing that “is a subset of” is transitive and “is asuperset of” is its converse, we can conclude that the latter is transitive as well.The intersection of two transitive relations is always transitive: knowing that “was born before” and “has the same firstname as” are transitive, we can conclude that “was born before and also has the same first name as” is also transitive.The union of two transitive relations is not always transitive. For instance “was born before or has the same first nameas” is not generally a transitive relation.The complement of a transitive relation is not always transitive. For instance, while “equal to” is transitive, “not equalto” is only transitive on sets with at most one element.

22.3.2 Other properties

A transitive relation is asymmetric if and only if it is irreflexive.[1]

22.3.3 Properties that require transitivity

• Preorder – a reflexive transitive relation

• partial order – an antisymmetric preorder

• Total preorder – a total preorder

• Equivalence relation – a symmetric preorder

• Strict weak ordering – a strict partial order in which incomparability is an equivalence relation

• Total ordering – a total, antisymmetric transitive relation

22.4 Counting transitive relations

No general formula that counts the number of transitive relations on a finite set (sequence A006905 in OEIS) isknown.[2] However, there is a formula for finding the number of relations that are simultaneously reflexive, symmet-ric, and transitive – in other words, equivalence relations – (sequence A000110 in OEIS), those that are symmetricand transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and an-tisymmetric. Pfeiffer[3] has made some progress in this direction, expressing relations with combinations of theseproperties in terms of each other, but still calculating any one is difficult. See also.[4]

22.5 See also

• Transitive closure

• Transitive reduction

• Intransitivity

• Reflexive relation

• Symmetric relation

• Quasitransitive relation

• Nontransitive dice

• Rational choice theory

22.6. SOURCES 61

22.6 Sources

22.6.1 References[1] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School

of Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.

[2] Steven R. Finch, “Transitive relations, topologies and partial orders”, 2003.

[3] Götz Pfeiffer, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

[4] Gunnar Brinkmann and Brendan D. McKay,”Counting unlabelled topologies and transitive relations"

22.6.2 Bibliography

• Ralph P. Grimaldi, Discrete and Combinatorial Mathematics, ISBN 0-201-19912-2.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

22.7 External links• Hazewinkel, Michiel, ed. (2001), “Transitivity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

• Transitivity in Action at cut-the-knot

Chapter 23

Trichotomy (mathematics)

In mathematics, the Law of Trichotomy states that every real number is either positive, negative, or zero.[1] Moregenerally, trichotomy is the property of an order relation < on a setX that for any x and y, exactly one of the followingholds: x < y , x = y , or x > y .In mathematical notation, this is

∀x ∈ X ∀y ∈ X ((x < y ∧¬(y < x)∧¬(x = y) )∨ (¬(x < y)∧ y < x∧¬(x = y) )∨ (¬(x < y)∧¬(y < x)∧x = y )) .

Assuming that the ordering is irreflexive and transitive, this can be simplified to

∀x ∈ X ∀y ∈ X ((x < y) ∨ (y < x) ∨ (x = y)) .

In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore alsofor comparisons between integers and between rational numbers. The law does not hold in general in intuitionisticlogic.In ZF set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers of well-orderablesets even without the axiom of choice. If the axiom of choice holds, then trichotomy holds between arbitrary cardinalnumbers (because they are all well-orderable in that case).[2]

More generally, a binary relation R on X is trichotomous if for all x and y in X exactly one of xRy, yRx or x=y holds.If such a relation is also transitive it is a strict total order; this is a special case of a strict weak order. For example,in the case of three element set {a,b,c} the relation R given by aRb, aRc, bRc is a strict total order, while the relationR given by the cyclic aRb, bRc, cRa is a non-transitive trichotomous relation.In the definition of an ordered integral domain or ordered field, the law of trichotomy is usually taken as morefoundational than the law of total order.A trichotomous relation cannot be reflexive, since xRx must be false. If a trichotomous relation is transitive, it istrivially antisymmetric and also asymmetric, since xRy and yRx cannot both hold.

23.1 See also• Dichotomy

• Law of noncontradiction

• Law of excluded middle

23.2 References[1] http://mathworld.wolfram.com/TrichotomyLaw.html

62

23.2. REFERENCES 63

[2] Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.

Chapter 24

Unimodality

“Unimodal” redirects here. For the company that promotes personal rapid transit, see SkyTran.

In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only asingle highest value, somehow defined, of some mathematical object.[1]

24.1 Unimodal probability distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-5 -4 -3 -2 -1 0 1 2 3 4 5

μ = 0, σ² = 0.2μ = 0, σ² = 1.0μ = 0, σ² = 5.0

μ = -2, σ² = 0.5

Figure 1. probability density function of normal distributions, an example of unimodal distribution.

In statistics, a unimodal probability distribution (or when referring to the distribution, a unimodal distribution)

64

24.1. UNIMODAL PROBABILITY DISTRIBUTION 65

Figure 2. a simple bimodal distribution.

Figure 3. a distribution which, though strictly unimodal, is usually referred to as bimodal.

is a probability distribution which has a single mode. As the term “mode” has multiple meanings, so does the term“unimodal”.Strictly speaking, a mode of a discrete probability distribution is a value at which the probability mass function (pmf)takes its maximum value. In other words, it is a most likely value. A mode of a continuous probability distributionis a value at which the probability density function (pdf) attains its maximum value. Note that in both cases therecan be more than one mode, since the maximum value of either the pmf or the pdf can be attained at more than onevalue.If there is a single mode, the distribution function is called “unimodal”. If it has more modes it is “bimodal” (2), “tri-modal” (3), etc., or in general, “multimodal”.[2] Figure 1 illustrates normal distributions, which are unimodal. Otherexamples of unimodal distributions include Cauchy distribution, Student’s t-distribution and chi-squared distribution.

66 CHAPTER 24. UNIMODALITY

Figure 2 illustrates a bimodal distribution.Figure 3 illustrates a distribution which by strict definition is unimodal. However, confusingly, and mostly withcontinuous distributions, when a pdf function has multiple local maxima it is common to refer to all of the localmaxima as modes of the distribution. Therefore, if a pdf has more than one local maximum it is referred to asmultimodal. Under this common definition, Figure 3 illustrates a bimodal distribution.

24.1.1 Other definitions

Other definitions of unimodality in distribution functions also exist.In continuous distributions, unimodality can be defined through the behavior of the cumulative distribution function(cdf).[3] If the cdf is convex for x < m and concave for x > m, then the distribution is unimodal, m being the mode.Note that under this definition the uniform distribution is unimodal,[4] as well as any other distribution in which themaximum distribution is achieved for a range of values, e.g. trapezoidal distribution. Note also that usually thisdefinition allows for a discontinuity at the mode; usually in a continuous distribution the probability of any singlevalue is zero, while this definition allows for a non-zero probability, or an “atom of probability”, at the mode.Criteria for unimodality can also be defined through the characteristic function of the distribution[3] or through itsLaplace–Stieltjes transform.[5]

Another way to define a unimodal discrete distribution is by the occurrence of sign changes in the sequence of dif-ferences of the probabilities.[6] A discrete distribution with a probability mass function, {pn;n = . . . ,−1, 0, 1, . . . }, is called unimodal if the sequence . . . , p−2 − p−1, p−1 − p0, p0 − p1, p1 − p2, . . . has exactly one sign change(when zeroes don't count).

24.1.2 Uses and results

One reason for the importance of distribution unimodality is that it allows for several important results. SeveralInequalities are given below which are only valid for unimodal distributions. Thus, it is important to assess whetheror not a given data set comes from a unimodal distribution. Several tests for unimodality are given in the Wiki articleon Multimodal Distribution

24.1.3 Gauss’ inequality

A first important result is Gauss’s inequality.[7] Gauss’s inequality gives an upper bound on the probability that a valuelies more than any given distance from its mode. This inequality depends on unimodality.

24.1.4 Vysochanskiï–Petunin inequality

A second is the Vysochanskiï–Petunin inequality,[8] a refinement of the Chebyshev inequality. The Chebyshev in-equality guarantees that in any probability distribution, “nearly all” the values are “close to” the mean value. TheVysochanskiï–Petunin inequality refines this to even nearer values, provided that the distribution function is uni-modal. Further results were shown by Sellke & Sellke.[9]

24.1.5 Mode, median and mean

It can be shown for a unimodal distribution that the median X̃ and the mean X̄ lie within (3/5)1/2 ≈ 0.7746 standarddeviations of each other.[10] In symbols,

∣∣∣X̃ − X̄∣∣∣

σ≤ (3/5)1/2

where |.| is the absolute value.A similar relation holds between the median and the mode: they lie within 31/2 ≈ 1.732 standard deviations of eachother:

24.2. UNIMODAL FUNCTION 67

∣∣∣X̃ − mode∣∣∣

σ≤ 31/2.

24.1.6 Skewness and kurtosis

Rohatgi and Szekely have shown that the skewness and kurtosis of a unimodal distribution are related by the inequality:[11]

γ2 − κ ≤ 6

5

where κ is the kurtosis and γ is the skewness.Klaassen, Mokveld, and van Es derived a slightly different inequality (shown below) from the one derived by Rohatgiand Szekely (shown above), which tends to be more inclusive (i.e., yield more positives) in tests of unimodality:[12]

γ2 − κ ≤ 186

125

24.2 Unimodal function

As the term “modal” applies to data sets and probability distribution, and not in general to functions, the definitionsabove do not apply. The definition of “unimodal” was extended to functions of real numbers as well.A common definition is as follows: a function f(x) is a unimodal function if for some value m, it is monotonicallyincreasing for x ≤ m and monotonically decreasing for x ≥ m. In that case, the maximum value of f(x) is f(m) andthere are no other local maxima.Proving unimodality is often hard. One way consists in using the definition of that property, but it turns out to besuitable for simple functions only. A general method based on derivatives exists,[13] but it does not succeed for everyfunction despite its simplicity.Examples of unimodal functions include quadratic polynomial functions with a negative quadratic coefficient, tentmap functions, and more.The above is sometimes related to as “strong unimodality”, from the fact that the monotonicity implied is strongmonotonicity. A function f(x) is a weakly unimodal function if there exists a value m for which it is weakly mono-tonically increasing for x ≤ m and weakly monotonically decreasing for x ≥ m. In that case, the maximum valuef(m) can be reached for a continuous range of values of x. An example of a weakly unimodal function which is notstrongly unimodal is every other row in a Pascal triangle.Depending on context, unimodal function may also refer to a function that has only one local minimum, rather thanmaximum.[14] For example, local unimodal sampling, a method for doing numerical optimization, is often demon-strated with such a function. It can be said that a unimodal function under this extension is a function with a singlelocal extremum.One important property of unimodal functions is that the extremum can be found using search algorithms such asgolden section search, ternary search or successive parabolic interpolation.

24.3 Other extensions

A function f(x) is “S-unimodal” (often referred to as “S-unimodal map”) if its Schwarzian derivative is negative forall x ̸= c , where c is the critical point.[15]

In computational geometry if a function is unimodal it permits the design of efficient algorithms for finding theextrema of the function.[16]

68 CHAPTER 24. UNIMODALITY

A more general definition, applicable to a function f(X) of a vector variable X is that f is unimodal if there is a one toone differentiable mapping X = G(Z) such that f(G(Z)) is convex. Usually one would want G(Z) to be continuouslydifferentiable with nonsingular Jacobian matrix.Quasiconvex functions and quasiconcave functions extend the concept of unimodality to functions whose argumentsbelong to higher-dimensional Euclidean spaces.

24.4 See also• Bimodal distribution

24.5 References[1] Weisstein, Eric W., “Unimodal”, MathWorld.

[2] Weisstein, Eric W., “Mode”, MathWorld.

[3] A.Ya. Khinchin (1938). “On unimodal distributions”. Trams. Res. Inst. Math. Mech. (in Russian) (University of Tomsk)2 (2): 1–7.

[4] Ushakov, N.G. (2001), “Unimodal distribution”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

[5] Vladimirovich Gnedenko and Victor Yu Korolev (1996). Random summation: limit theorems and applications. CRC-Press.ISBN 0-8493-2875-6. p. 31

[6] Medgyessy, P. (March 1972). “On the unimodality of discrete distributions”. Periodica Mathematica Hungarica 2 (1–4):245–257. doi:10.1007/bf02018665.

[7] Gauss, C. F. (1823). “Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Pars Prior”. CommentationesSocietatis Regiae Scientiarum Gottingensis Recentiores 5.

[8] D. F. Vysochanskij, Y. I. Petunin (1980). “Justification of the 3σ rule for unimodal distributions”. Theory of Probabilityand Mathematical Statistics 21: 25–36.

[9] Sellke, T.M.; Sellke, S.H. (1997). “Chebyshev inequalities for unimodal distributions”. American Statistician (AmericanStatistical Association) 51 (1): 34–40. doi:10.2307/2684690. JSTOR 2684690.

[10] Basu, Sanjib, and Anirban DasGupta. “The mean, median, and mode of unimodal distributions: a characterization.” Theoryof Probability & Its Applications 41.2 (1997): 210-223.

[11] Rohatgi VK, Szekely GJ (1989) Sharp inequalities between skewness and kurtosis. Statistics & Probability Letters 8:297-299

[12] Klaassen CAJ, Mokveld PJ, van Es B (2000) Squared skewness minus kurtosis bounded by 186/125 for unimodal distri-butions. Stat & Prob Lett 50 (2) 131–135

[13] “On the unimodality of METRIC Approximation subject to normally distributed demands.” (PDF). Method in appendixD, Example in theorem 2 page 5. Retrieved 2013-08-28.

[14] “Mathematical Programming Glossary.”. Retrieved 2010-07-07.

[15] See e.g. John Guckenheimer and Stewart Johnson (July 1990). “Distortion of S-Unimodal Maps”. The Annals of Mathe-matics, Second Series 132 (1). pp. 71–130. doi:10.2307/1971501.

[16] Godfried T. Toussaint (June 1984). “Complexity, convexity, and unimodality”. International Journal of Computer andInformation Sciences 13 (3). pp. 197–217. doi:10.1007/bf00979872.

Chapter 25

Weak ordering

Not to be confused with weak order of permutations.In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of

a<bc<b

a<ba<c

b<ac<a

b<ab<c

c<ac<b

a<cb<ca,b,c

c<b<a

b<c<a

b<a<c

c<a<b a<b<c

a<c<b

The 13 possible strict weak orderings on a set of three elements {a, b, c}. The only partially ordered sets are coloured, while totallyordered ones are in black. Two orderings are shown as connected by an edge if they differ by a single dichotomy.

69

70 CHAPTER 25. WEAK ORDERING

a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totallyordered sets (rankings without ties) and are in turn generalized by partially ordered sets and preorders.[1]

There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic(interconvertable with no loss of information): they may be axiomatized as strict weak orderings (partially orderedsets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at leastone of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of theelements into disjoint subsets, together with a total order on the subsets). In many cases another representation calleda preferential arrangement based on a utility function is also possible.Weak orderings are counted by the ordered Bell numbers. They are used in computer science as part of partitionrefinement algorithms, and in the C++ Standard Library.

25.1 Examples

In horse racing, the use of photo finishes has eliminated some, but not all, ties or (as they are called in this context)dead heats, so the outcome of a horse race may be modeled by a weak ordering.[2] In an example from the MarylandHunt Cup steeplechase in 2007, The Bruce was the clear winner, but two horses Bug River and Lear Charm tied forsecond place, with the remaining horses farther back; three horses did not finish.[3] In the weak ordering describingthis outcome, The Bruce would be first, Bug River and Lear Charm would be ranked after The Bruce but before allthe other horses that finished, and the three horses that did not finish would be placed last in the order but tied witheach other.The points of the Euclidean plane may be ordered by their distance from the origin, giving another example of a weakordering with infinitely many elements, infinitely many subsets of tied elements (the sets of points that belong to acommon circle centered at the origin), and infinitely many points within these subsets. Although this ordering has asmallest element (the origin itself), it does not have any second-smallest elements, nor any largest element.Opinion polling in political elections provides an example of a type of ordering that resembles weak orderings, but isbetter modeled mathematically in other ways. In the results of a poll, one candidate may be clearly ahead of another,or the two candidates may be statistically tied, meaning not that their poll results are equal but rather that they arewithin the margin of error of each other. However, if candidate x is statistically tied with y, and y is statistically tiedwith z, it might still be possible for x to be clearly better than z, so being tied is not in this case a transitive relation.Because of this possibility, rankings of this type are better modeled as semiorders than as weak orderings.[4]

25.2 Axiomatizations

25.2.1 Strict weak orderings

A strict weak ordering is a binary relation < on a set S that is a strict partial order (a transitive relation that isirreflexive, or equivalently,[5] that is asymmetric) in which the relation “neither a < b nor b < a" is transitive.[1]

The equivalence classes of this “incomparability relation” partition the elements of S, and are totally ordered by <.Conversely, any total order on a partition of S gives rise to a strict weak ordering in which x < y if and only if thereexists sets A and B in the partition with x in A, y in B, and A < B in the total order.As a non-example, consider the partial order in the set {a, b, c} defined by the relationship b < c. The pairs a,b and a,care incomparable but b and c are related, so incomparability does not form an equivalence relation and this exampleis not a strict weak ordering.A strict weak ordering has the following properties. For all x and y in S,

• For all x, it is not the case that x < x (irreflexivity).

• For all x, y, if x < y then it is not the case that y < x (asymmetry).

• For all x, y, and z, if x < y and y < z then x < z (transitivity).

• For all x, y, and z, if x is incomparable with y, and y is incomparable with z, then x is incomparable with z(transitivity of incomparability).

25.2. AXIOMATIZATIONS 71

This list of properties is somewhat redundant, as asymmetry follows readily from irreflexivity and transitivity.Transitivity of incomparability (together with transitivity) can also be stated in the following forms:

• If x < y, then for all z, either x < z or z < y or both.

Or:

• If x is incomparable with y, then for all z ≠ x, z ≠ y, either (x < z and y < z) or (z < x and z < y) or (z isincomparable with x and z is incomparable with y).

25.2.2 Total preorders

Strict weak orders are very closely related to total preorders or (non-strict) weak orders, and the same mathematicalconcepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A totalpreorder or weak order is a preorder that is total; that is, no pair of items is incomparable. A total preorder ≲ satisfiesthe following properties:

• For all x, y, and z, if x ≲ y and y ≲ z then x ≲ z (transitivity).

• For all x and y, x ≲ y or y ≲ x (totality).

• Hence, for all x, x ≲ x (reflexivity).

A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preordersare sometimes also called preference relations.The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strictweak orders and total preorders in a way that preserves rather than reverses the order of the elements. Thus we takethe inverse of the complement: for a strict weak ordering <, define a total preorder ≲ by setting x ≲ y whenever it isnot the case that y < x. In the other direction, to define a strict weak ordering < from a total preorder ≲ , set x < ywhenever it is not the case that y ≲ x.[6]

In any preorder there is a corresponding equivalence relation where two elements x and y are defined as equivalent ifx ≲ y and y ≲ x. In the case of a total preorder the corresponding partial order on the set of equivalence classes is atotal order. Two elements are equivalent in a total preorder if and only if they are incomparable in the correspondingstrict weak ordering.

25.2.3 Ordered partitions

A partition of a set S is a family of disjoint subsets of S that have S as their union. A partition, together with atotal order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition[7] and byTheodore Motzkin a list of sets.[8] An ordered partition of a finite set may be written as a finite sequence of the setsin the partition: for instance, the three ordered partitions of the set {a, b} are

{a}, {b},{b}, {a}, and{a, b}.

In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit atotal ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partitiongives rise to a strict weak ordering in which two elements are incomparable when they belong to the same set in thepartition, and otherwise inherit the order of the sets that contain them.

72 CHAPTER 25. WEAK ORDERING

25.2.4 Representation by functions

For sets of sufficiently small cardinality, a third axiomatization is possible, based on real-valued functions. If X is anyset and f a real-valued function on X then f induces a strict weak order on X by setting a < b if and only if f(a) < f(b).The associated total preorder is given by setting a ≲ b if and only if f(a) ≤ f(b), and the associated equivalence bysetting a ∼ b if and only if f(a) = f(b).The relations do not change when f is replaced by g o f (composite function), where g is a strictly increasing real-valued function defined on at least the range of f. Thus e.g. a utility function defines a preference relation. In thiscontext, weak orderings are also known as preferential arrangements.[9]

If X is finite or countable, every weak order on X can be represented by a function in this way.[10] However, thereexist strict weak orders that have no corresponding real function. For example, there is no such function for thelexicographic order on Rn. Thus, while in most preference relation models the relation defines a utility function up toorder-preserving transformations, there is no such function for lexicographic preferences.More generally, if X is a set, and Y is a set with a strict weak ordering "<", and f a function from X to Y, then finduces a strict weak ordering on X by setting a < b if and only if f(a) < f(b). As before, the associated total preorderis given by setting a ≲ b if and only if f(a) ≲ f(b), and the associated equivalence by setting a ∼ b if and onlyif f(a) ∼ f(b). It is not assumed here that f is an injective function, so a class of two equivalent elements on Ymay induce a larger class of equivalent elements on X. Also, f is not assumed to be an surjective function, so a classof equivalent elements on Y may induce a smaller or empty class on X. However, the function f induces an injectivefunction that maps the partition on X to that on Y. Thus, in the case of finite partitions, the number of classes in X isless than or equal to the number of classes on Y.

25.3 Related types of ordering

Semiorders generalize strict weak orderings, but do not assume transitivity of incomparability.[11] A strict weak orderthat is trichotomous is called a strict total order.[12] The total preorder which is the inverse of its complement is inthis case a total order.For a strict weak order "<" another associated reflexive relation is its reflexive closure, a (non-strict) partial order "≤".The two associated reflexive relations differ with regard to different a and b for which neither a < b nor b < a: in thetotal preorder corresponding to a strict weak order we get a ≲ b and b ≲ a, while in the partial order given by thereflexive closure we get neither a ≤ b nor b ≤ a. For strict total orders these two associated reflexive relations arethe same: the corresponding (non-strict) total order.[12] The reflexive closure of a strict weak ordering is a type ofseries-parallel partial order.

25.4 All weak orders on a finite set

25.4.1 Combinatorial enumeration

Main article: ordered Bell number

The number of distinct weak orders (represented either as strict weak orders or as total preorders) on an n-elementset is given by the following sequence (sequence A000670 in OEIS):These numbers are also called the Fubini numbers or ordered Bell numbers.For example, for a set of three labeled items, there is one weak order in which all three items are tied. There are threeways of partitioning the items into one singleton set and one group of two tied items, and each of these partitionsgives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering isreversed), giving six weak orders of this type. And there is a single way of partitioning the set into three singletons,which can be totally ordered in six different ways. Thus, altogether, there are 13 different weak orders on three items.

25.4. ALL WEAK ORDERS ON A FINITE SET 73

(4,1,2,3)(4,2,1,3)

(3,2,1,4)

(3,1,2,4)

(2,1,3,4)

(1,2,3,4)

(1,2,4,3)

(1,3,2,4)

(2,1,4,3)

(2,3,1,4)

(3,1,4,2)

(4,1,3,2)

(4,2,3,1)

(3,2,4,1)(2,4,1,3)

(1,4,2,3)

(1,3,4,2)

(2,3,4,1)

(1,4,3,2)

(2,4,3,1)

(3,4,2,1)

(4,3,2,1)

(4,3,1,2)

(3,4,1,2)

The permutohedron on four elements, a three-dimensional convex polyhedron. It has 24 vertices, 36 edges, and 14 two-dimensionalfaces, which all together with the whole three-dimensional polyhedron correspond to the 75 weak orderings on four elements.

25.4.2 Adjacency structure

Unlike for partial orders, the family of weak orderings on a given finite set is not in general connected by moves thatadd or remove a single order relation to a given ordering. For instance, for three elements, the ordering in which allthree elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strictweak ordering or total preorder axiomatizations. However, a different kind of move is possible, in which the weakorderings on a set are more highly connected. Define a dichotomy to be a weak ordering with two equivalence classes,and define a dichotomy to be compatible with a given weak ordering if every two elements that are related in theordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as aDedekind cut for a weak ordering. Then a weak ordering may be characterized by its set of compatible dichotomies.For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of movesthat add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the undirected graph thathas the weak orderings as its vertices, and these moves as its edges, forms a partial cube.[13]

Geometrically, the total orderings of a given finite set may be represented as the vertices of a permutohedron, and thedichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderingson the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedronitself, but not the empty set, as a face). The codimension of a face gives the number of equivalence classes in thecorresponding weak ordering.[14] In this geometric representation the partial cube of moves on weak orderings is thegraph describing the covering relation of the face lattice of the permutohedron.For instance, for n = 3, the permutohedron on three elements is just a regular hexagon. The face lattice of the hexagon(again, including the hexagon itself as a face, but not including the empty set) has thirteen elements: one hexagon,six edges, and six vertices, corresponding to the one completely tied weak ordering, six weak orderings with one tie,and six total orderings. The graph of moves on these 13 weak orderings is shown in the figure.

74 CHAPTER 25. WEAK ORDERING

25.5 Applications

As mentioned above, weak orders have applications in utility theory.[10] In linear programming and other types ofcombinatorial optimization problem, the prioritization of solutions or of bases is often given by a weak order, de-termined by a real-valued objective function; the phenomenon of ties in these orderings is called “degeneracy”, andseveral types of tie-breaking rule have been used to refine this weak ordering into a total ordering in order to preventproblems caused by degeneracy.[15]

Weak orders have also been used in computer science, in partition refinement based algorithms for lexicographicbreadth-first search and lexicographic topological ordering. In these algorithms, a weak ordering on the vertices of agraph (represented as a family of sets that partition the vertices, together with a doubly linked list providing a totalorder on the sets) is gradually refined over the course of the algorithm, eventually producing a total ordering that isthe output of the algorithm.[16]

In the Standard Library for the C++ programming language, the set and multiset data types sort their input by acomparison function that is specified at the time of template instantiation, and that is assumed to implement a strictweak ordering.[17]

25.6 References[1] Roberts, Fred; Tesman, Barry (2011), Applied Combinatorics (2nd ed.), CRC Press, Section 4.2.4 Weak Orders, pp. 254–

256, ISBN 9781420099836.

[2] de Koninck, J. M. (2009), Those Fascinating Numbers, American Mathematical Society, p. 4, ISBN 9780821886311.

[3] Baker, Kent (April 29, 2007), “The Bruce hangs on for Hunt Cup victory: Bug River, Lear Charm finish in dead heat forsecond”, The Baltimore Sun, (subscription required (help)).

[4] Regenwetter, Michel (2006), Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications, Cam-bridge University Press, pp. 113ff, ISBN 9780521536660.

[5] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.

[6] Ehrgott, Matthias (2005), Multicriteria Optimization, Springer, Proposition 1.9, p. 10, ISBN 9783540276593.

[7] Stanley, Richard P. (1997), Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cam-bridge University Press, p. 297.

[8] Motzkin, Theodore S. (1971), “Sorting numbers for cylinders and other classification numbers”, Combinatorics (Proc.Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968), Providence, R.I.: Amer. Math. Soc., pp.167–176, MR 0332508.

[9] Gross, O. A. (1962), “Preferential arrangements”, The American Mathematical Monthly 69: 4–8, doi:10.2307/2312725,MR 0130837.

[10] Roberts, Fred S. (1979), Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences, Ency-clopedia of Mathematics and its Applications 7, Addison-Wesley, Theorem 3.1, ISBN 978-0-201-13506-0.

[11] Luce, R. Duncan (1956), “Semiorders and a theory of utility discrimination”, Econometrica 24: 178–191, JSTOR 1905751,MR 0078632.

[12] Velleman, Daniel J. (2006),How to Prove It: A Structured Approach, Cambridge University Press, p. 204, ISBN 9780521675994.

[13] Eppstein, David; Falmagne, Jean-Claude; Ovchinnikov, Sergei (2008), Media Theory: Interdisciplinary Applied Mathemat-ics, Springer, Section 9.4, Weak Orders and Cubical Complexes, pp. 188–196.

[14] Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, p. 18.

[15] Chvátal, Vašek (1983), Linear Programming, Macmillan, pp. 29–38, ISBN 9780716715870.

[16] Habib, Michel; Paul, Christophe; Viennot, Laurent (1999), “Partition refinement techniques: an interesting algorithmictool kit”, International Journal of Foundations of Computer Science 10 (2): 147–170, doi:10.1142/S0129054199000125,MR 1759929.

[17] Josuttis, Nicolai M. (2012), The C++ Standard Library: A Tutorial and Reference, Addison-Wesley, p. 469, ISBN9780132977739.

Chapter 26

Well-founded relation

“Noetherian induction” redirects here. For the use in topology, see Noetherian topological space.

In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class X if and only if every non-emptysubset S X has a minimal element; that is, some element m of any S is not related by sRm (for instance, "m is notsmaller than”) for the rest of the s ∈ S.

∀S ⊆ X (S ̸= ∅ → ∃m ∈ S ∀s ∈ S (s,m) /∈ R)

(Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form aset.)Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descend-ing chains: that is, there is no infinite sequence x0, x1, x2, ... of elements of X such that xn₊₁ R x for every naturalnumber n.In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. Ifthe order is a total order then it is called a well-order.In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitiveclosure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all setsare well-founded.A relation R is converse well-founded, upwards well-founded or Noetherian on X, if the converse relation R−1

is well-founded on X. In this case R is also said to satisfy the ascending chain condition. In the context of rewritingsystems, a Noetherian relation is also called terminating.

26.1 Induction and recursion

An important reason that well-founded relations are interesting is because a version of transfinite induction can beused on them: if (X, R) is a well-founded relation, P(x) is some property of elements of X, and we want to show that

P(x) holds for all elements x of X,

it suffices to show that:

If x is an element of X and P(y) is true for all y such that y R x, then P(x) must also be true.

That is,

∀x ∈ X [(∀y ∈ X (y Rx → P (y))) → P (x)] → ∀x ∈ X P (x).

75

76 CHAPTER 26. WELL-FOUNDED RELATION

Well-founded induction is sometimes called Noetherian induction,[1] after Emmy Noether.On par with induction, well-founded relations also support construction of objects by transfinite recursion. Let (X,R) be a set-like well-founded relation and F a function that assigns an object F(x, g) to each pair of an element x ∈ Xand a function g on the initial segment {y: y R x} of X. Then there is a unique function G such that for every x ∈ X,

G(x) = F (x,G|{y:y Rx})

That is, if we want to construct a function G on X, we may define G(x) using the values of G(y) for y R x.As an example, consider the well-founded relation (N, S), where N is the set of all natural numbers, and S is the graphof the successor function x → x + 1. Then induction on S is the usual mathematical induction, and recursion on Sgives primitive recursion. If we consider the order relation (N, <), we obtain complete induction, and course-of-valuesrecursion. The statement that (N, <) is well-founded is also known as the well-ordering principle.There are other interesting special cases of well-founded induction. When the well-founded relation is the usualordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded setis a set of recursively-defined data structures, the technique is called structural induction. When the well-foundedrelation is set membership on the universal class, the technique is known as ∈-induction. See those articles for moredetails.

26.2 Examples

Well-founded relations which are not totally ordered include:

• the positive integers {1, 2, 3, ...}, with the order defined by a < b if and only if a divides b and a ≠ b.

• the set of all finite strings over a fixed alphabet, with the order defined by s < t if and only if s is a propersubstring of t.

• the set N × N of pairs of natural numbers, ordered by (n1, n2) < (m1, m2) if and only if n1 < m1 and n2 < m2.

• the set of all regular expressions over a fixed alphabet, with the order defined by s < t if and only if s is a propersubexpression of t.

• any class whose elements are sets, with the relation ∈ (“is an element of”). This is the axiom of regularity.

• the nodes of any finite directed acyclic graph, with the relation R defined such that a R b if and only if there isan edge from a to b.

Examples of relations that are not well-founded include:

• the negative integers {−1, −2, −3, …}, with the usual order, since any unbounded subset has no least element.

• The set of strings over a finite alphabet with more than one element, under the usual (lexicographic) order,since the sequence “B” > “AB” > “AAB” > “AAAB” > … is an infinite descending chain. This relation failsto be well-founded even though the entire set has a minimum element, namely the empty string.

• the rational numbers (or reals) under the standard ordering, since, for example, the set of positive rationals (orreals) lacks a minimum.

26.3 Other properties

If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, butthis does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union ofthe positive integers and a new element ω, which is bigger than any integer. Then X is a well-founded set, but thereare descending chains starting at ω of arbitrary great (finite) length; the chain ω, n − 1, n − 2, ..., 2, 1 has length nfor any n.

26.4. REFLEXIVITY 77

The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded rela-tions: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X,R)is isomorphic to (C,∈).

26.4 Reflexivity

A relation R is said to be reflexive if a R a holds for every a in the domain of the relation. Every reflexive relation on anonempty domain has infinite descending chains, because any constant sequence is a descending chain. For example,in the natural numbers with their usual order ≤, we have 1 ≥ 1 ≥ 1 ≥ · · · . To avoid these trivial descendingsequences, when working with a reflexive relation R it is common to use (perhaps implicitly) the alternate relation R′defined such that a R′ b if and only if a R b and a ≠ b. In the context of the natural numbers, this means that therelation <, which is well-founded, is used instead of the relation ≤, which is not. In some texts, the definition of awell-founded relation is changed from the definition above to include this convention.

26.5 References[1] Bourbaki, N. (1972) Elements of mathematics. Commutative algebra, Addison-Wesley.

• Just, Winfried and Weese, Martin, Discovering Modern Set theory. I, American Mathematical Society (1998)ISBN 0-8218-0266-6.

Chapter 27

Well-order

In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that everynon-empty subset of S has a least element in this ordering. The set S together with the well-order relation is thencalled awell-ordered set. The hyphen is frequently omitted in contemporary papers, yielding the spellingswellorder,wellordered, and wellordering.Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatestelement, has a unique successor (next element), namely the least element of the subset of all elements greater thans. There may be elements besides the least element which have no predecessor (see Natural numbers below for anexample). In a well-ordered set S, every subset T which has an upper bound has a least upper bound, namely the leastelement of the subset of all upper bounds of T in S.If ≤ is a non-strict well-ordering, then < is a strict well-ordering. A relation is a strict well-ordering if and only if it isa well-founded strict total order. The distinction between strict and non-strict well-orders is often ignored since theyare easily interconvertible.Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can bewell-ordered. If a set is well-ordered (or even if it merely admits a wellfounded relation), the proof technique oftransfinite induction can be used to prove that a given statement is true for all elements of the set.The observation that the natural numbers are well-ordered by the usual less-than relation is commonly called thewell-ordering principle (for natural numbers).

27.1 Ordinal numbers

Main article: Ordinal number

Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of afinite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object witha particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (numberof elements, cardinal number) of a finite set is equal to the order type. Counting in the everyday sense typically startsfrom one, so it assigns to each object the size of the initial segment with that object as last element. Note that thesenumbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equalto the number of earlier objects (which corresponds to counting from zero). Thus for finite n, the expression "n-thelement” of a well-ordered set requires context to know whether this counts from zero or one. In a notation "β-thelement” where β can also be an infinite ordinal, it will typically count from zero.For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particularcardinality can have many different order types. For a countably infinite set, the set of possible order types is evenuncountable.

78

27.2. EXAMPLES AND COUNTEREXAMPLES 79

27.2 Examples and counterexamples

27.2.1 Natural numbers

The standard ordering ≤ of the natural numbers is a well-ordering and has the additional property that every non-zeronatural number has a unique predecessor.Another well-ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers,and the usual ordering applies within the evens and the odds:

0 2 4 6 8 ... 1 3 5 7 9 ...

This is a well-ordered set of order type ω + ω. Every element has a successor (there is no largest element). Twoelements lack a predecessor: 0 and 1.

27.2.2 Integers

Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well-ordering,since, for example, the set of negative integers does not contain a least element.The following relation R is an example of well-ordering of the integers: x R y if and only if one of the followingconditions holds:

1. x = 0

2. x is positive, and y is negative

3. x and y are both positive, and x ≤ y

4. x and y are both negative, and |x| ≤ |y|

This relation R can be visualized as follows:

0 1 2 3 4 ... −1 −2 −3 ...

R is isomorphic to the ordinal number ω + ω.Another relation for well-ordering the integers is the following definition: x ≤ y iff (|x| < |y| or (|x| = |y| and x ≤ y)).This well-order can be visualized as follows:

0 −1 1 −2 2 −3 3 −4 4 ...

This has the order type ω.

27.2.3 Reals

The standard ordering ≤ of the positive real numbers is not a well-ordering, since, for example, the open interval (0,1) does not contain a least element. From the ZFC axioms of set theory (including the axiom of choice) one can showthat there is a well-order of the reals. Also Wacław Sierpiński proved that ZF + GCH (the generalized continuumhypothesis) imply the axiom of choice and hence a well-order of the reals. Nonetheless, it is possible to show thatthe ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well-order ofthe reals.[1] However it is consistent with ZFC that a definable well-ordering of the reals exists—for example, it isconsistent with ZFC that V=L, and it follows from ZFC+V=L that a particular formula well-orders the reals, or indeedany set.An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well-order: SupposeX is a subsetof R well-ordered by ≤. For each x in X, let s(x) be the successor of x in ≤ ordering on X (unless x is the last elementof X). Let A = { (x, s(x)) | x ∈ X } whose elements are nonempty and disjoint intervals. Each such interval contains

80 CHAPTER 27. WELL-ORDER

at least one rational number, so there is an injective function from A to Q. There is an injection from X to A (exceptpossibly for a last element of X which could be mapped to zero later). And it is well known that there is an injectionfrom Q to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from X to thenatural numbers which means that X is countable. On the other hand, a countably infinite subset of the reals may ormay not be a well-order with the standard "≤".

• The natural numbers are a well-order.• The set {1/n : n =1,2,3,...} has no least element and is therefore not a well-order.

Examples of well-orders:

• The set of numbers { − 2−n | 0 ≤ n < ω } has order type ω.• The set of numbers { − 2−n − 2−m−n | 0 ≤ m,n < ω } has order type ω². The previous set is the set of limit

points within the set. Within the set of real numbers, either with the ordinary topology or the order topology,0 is also a limit point of the set. It is also a limit point of the set of limit points.

• The set of numbers { − 2−n | 0 ≤ n < ω } ∪ { 1 } has order type ω + 1. With the order topology of this set, 1 isa limit point of the set. With the ordinary topology (or equivalently, the order topology) of the real numbers itis not.

27.3 Equivalent formulations

If a set is totally ordered, then the following are equivalent to each other:

1. The set is well-ordered. That is, every nonempty subset has a least element.2. Transfinite induction works for the entire ordered set.3. Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming

the axiom of dependent choice).4. Every subordering is isomorphic to an initial segment.

27.4 Order topology

Every well-ordered set can be made into a topological space by endowing it with the order topology.With respect to this topology there can be two kinds of elements:

• isolated points - these are the minimum and the elements with a predecessor.• limit points - this type does not occur in finite sets, and may or may not occur in an infinite set; the infinite sets

without limit point are the sets of order type ω, for example N.

For subsets we can distinguish:

• Subsets with a maximum (that is, subsets which are bounded by themselves); this can be an isolated point or alimit point of the whole set; in the latter case it may or may not be also a limit point of the subset.

• Subsets which are unbounded by themselves but bounded in the whole set; they have no maximum, but asupremum outside the subset; if the subset is non-empty this supremum is a limit point of the subset and hencealso of the whole set; if the subset is empty this supremum is the minimum of the whole set.

• Subsets which are unbounded in the whole set.

A subset is cofinal in the whole set if and only if it is unbounded in the whole set or it has a maximum which is alsomaximum of the whole set.A well-ordered set as topological space is a first-countable space if and only if it has order type less than or equal toω1 (omega-one), that is, if and only if the set is countable or has the smallest uncountable order type.

27.5. SEE ALSO 81

27.5 See also• Tree (set theory), generalization

• Well-ordering theorem

• Ordinal number

• Well-founded set

• Well partial order

• Prewellordering

• Directed set

27.6 References[1] S. Feferman: “Some Applications of the Notions of Forcing and Generic Sets”, Fundamenta Mathematicae, 56 (1964)

325-345

• Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications. Pure and applied math-ematics (2nd ed.). John Wiley & Sons. pp. 4–6, 9. ISBN 978-0-471-31716-6.

Chapter 28

Well-quasi-ordering

In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering that is well-founded,meaning that any infinite sequence of elements x0 , x1 , x2 , … from X contains an increasing pair xi ≤ xj withi < j .

28.1 Motivation

Well-founded induction can be used on any set with a well-founded relation, thus one is interested in when a quasi-order is well-founded. However the class of well-founded quasiorders is not closed under certain operations - thatis, when a quasi-order is used to obtain a new quasi-order on a set of structures derived from our original set, thisquasiorder is found to be not well-founded. By placing stronger restrictions on the original well-founded quasiorderingone can hope to ensure that our derived quasiorderings are still well-founded.An example of this is the power set operation. Given a quasiordering ≤ for a set X one can define a quasiorder ≤+

on X 's power set P (X) by setting A ≤+ B if and only if for each element of A one can find some element of Bwhich is larger than it under ≤ . One can show that this quasiordering on P (X) needn't be well-founded, but if onetakes the original quasi-ordering to be a well-quasi-ordering, then it is.

28.2 Formal definition

A well-quasi-ordering on a set X is a quasi-ordering (i.e., a reflexive, transitive binary relation) such that any infinitesequence of elements x0 , x1 , x2 , … from X contains an increasing pair xi ≤ xj with i < j . The set X is said tobe well-quasi-ordered, or shortly wqo.A well partial order, or a wpo, is a wqo that is a proper ordering relation, i.e., it is antisymmetric.Among other ways of defining wqo’s, one is to say that they are quasi-orderings which do not contain infinite strictlydecreasing sequences (of the form x0 > x1 > x2 >…) nor infinite sequences of pairwise incomparable elements.Hence a quasi-order ( X ,≤) is wqo if and only if it is well-founded and has no infinite antichains.

28.3 Examples

• (N,≤) , the set of natural numbers with standard ordering, is a well partial order (in fact, a well-order). How-ever, (Z,≤) , the set of positive and negative integers, is not a well-quasi-order, because it is not well-founded.

• (N, |) , the set of natural numbers ordered by divisibility, is not a well partial order: the prime numbers are aninfinite antichain.

• (Nk,≤) , the set of vectors of k natural numbers (where k is finite) with component-wise ordering, is a wellpartial order (Dickson’s lemma). More generally, if (X,≤) is well-quasi-order, then (Xk,≤k) is also a well-quasi-order for all k .

82

28.4. WQO’S VERSUS WELL PARTIAL ORDERS 83

• LetX be an arbitrary finite set with at least two elements. The setX∗ of words overX ordered lexicographically(as in a dictionary) isnot a well-quasi-order because it contains the infinite decreasing sequence b, ab, aab, aaab, . . .. Similarly, X∗ ordered by the prefix relation is not a well-quasi-order, because the previous sequence is an in-finite antichain of this partial order. However, X∗ ordered by the subsequence relation is a well partial order.[1]

(If X has only one element, these three partial orders are identical.)

• More generally, (X∗,≤) , the set of finite X -sequences ordered by embedding is a well-quasi-order if andonly if (X,≤) is a well-quasi-order (Higman’s lemma). Recall that one embeds a sequence u into a sequence vby finding a subsequence of v that has the same length as u and that dominates it term by term. When (X,=)is a finite unordered set, u ≤ v if and only if u is a subsequence of v .

• (Xω,≤) , the set of infinite sequences over a well-quasi-order (X,≤) , ordered by embedding, is not a well-quasi-order in general. That is, Higman’s lemma does not carry over to infinite sequences. Better-quasi-orderings have been introduced to generalize Higman’s lemma to sequences of arbitrary lengths.

• Embedding between finite trees with nodes labeled by elements of a wqo (X,≤) is a wqo (Kruskal’s treetheorem).

• Embedding between infinite trees with nodes labeled by elements of a wqo (X,≤) is a wqo (Nash-Williams'theorem).

• Embedding between countable scattered linear order types is a well-quasi-order (Laver's theorem).

• Embedding between countable boolean algebras is a well-quasi-order. This follows from Laver’s theorem anda theorem of Ketonen.

• Finite graphs ordered by a notion of embedding called "graph minor" is a well-quasi-order (Robertson–Seymourtheorem).

• Graphs of finite tree-depth ordered by the induced subgraph relation form a well-quasi-order,[2] as do thecographs ordered by induced subgraphs.[3]

28.4 Wqo’s versus well partial orders

In practice, the wqo’s one manipulates are quite often not orderings (see examples above), and the theory is technicallysmoother if we do not require antisymmetry, so it is built with wqo’s as the basic notion.Observe that a wpo is a wqo, and that a wqo gives rise to a wpo between equivalence classes induced by the kernelof the wqo. For example, if we order Z by divisibility, we end up with n ≡ m if and only if n = ±m , so that(Z, |) ≈ (N, |) .

28.5 Infinite increasing subsequences

If ( X , ≤) is wqo then every infinite sequence x0 , x1 , x2 , … contains an infinite increasing subsequence xn0 ≤xn1 ≤ xn2 ≤… (with n0 < n1 < n2 <…). Such a subsequence is sometimes called perfect. This can be proved bya Ramsey argument: given some sequence (xi)i , consider the set I of indexes i such that xi has no larger or equalxj to its right, i.e., with i < j . If I is infinite, then the I -extracted subsequence contradicts the assumption that Xis wqo. So I is finite, and any xn with n larger than any index in I can be used as the starting point of an infiniteincreasing subsequence.The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering,leading to an equivalent notion.

28.6 Properties of wqos

• Given a quasiordering (X,≤) the quasiordering (P (X),≤+) defined by A ≤+ B ⇐⇒ ∀a ∈ A∃b ∈ B(a ≤b) is well-founded if and only if (X,≤) is a wqo.[4]

84 CHAPTER 28. WELL-QUASI-ORDERING

• A quasiordering is a wqo if and only if the corresponding partial order (obtained by quotienting by x ∼ y ⇐⇒x ≤ y ∧ y ≤ x ) has no infinite descending sequences or antichains. (This can be proved using a Ramseyargument as above.)

• Given a well-quasi-ordering (X,≤) , any sequence of subsets S0 ⊆ S1 ⊆ ... ⊆ X such that ∀i ∈ N, ∀x, y ∈X,x ≤ y∧x ∈ Si ⇒ y ∈ Si eventually stabilises (meaning there is an indexn ∈ N such thatSn = Sn+1 = ...; subsets S ⊆ X with the property ∀x, y ∈ X,x ≤ y ∧ x ∈ S ⇒ y ∈ S are usually called upward-closed):assuming the contrary ∀i ∈ N∃j ∈ N, j > i, ∃x ∈ Sj \Si , a contradiction is reached by extracting an infinitenon-ascending subsequence.

• Given a well-quasi-ordering (X,≤) , any subset S ⊆ X which is upward-closed with respect to ≤ has a finitenumber of minimal elements w.r.t. ≤ , for otherwise the minimal elements of S would constitute an infiniteantichain.

28.7 Notes[1] Gasarch, W. (1998), “A survey of recursive combinatorics”, Handbook of Recursive Mathematics, Vol. 2, Stud. Logic

Found. Math. 139, Amsterdam: North-Holland, pp. 1041–1176, doi:10.1016/S0049-237X(98)80049-9, MR 1673598.See in particular page 1160.

[2] Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), “Lemma 6.13”, Sparsity: Graphs, Structures, and Algorithms, Algo-rithms and Combinatorics 28, Heidelberg: Springer, p. 137, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7,MR 2920058.

[3] Damaschke, Peter (1990), “Induced subgraphs and well-quasi-ordering”, Journal of Graph Theory 14 (4): 427–435,doi:10.1002/jgt.3190140406, MR 1067237.

[4] Forster, Thomas (2003). “Better-quasi-orderings and coinduction”. Theoretical Computer Science 309 (1–3): 111–123.doi:10.1016/S0304-3975(03)00131-2.

28.8 References• Dickson, L. E. (1913). “Finiteness of the odd perfect and primitive abundant numbers with r distinct prime

factors”. American Journal of Mathematics 35 (4): 413–422. doi:10.2307/2370405. JSTOR 2370405.

• Higman, G. (1952). “Ordering by divisibility in abstract algebras”. Proceedings of the London MathematicalSociety 2: 326–336. doi:10.1112/plms/s3-2.1.326.

• Kruskal, J. B. (1972). “The theory of well-quasi-ordering: A frequently discovered concept”. Journal ofCombinatorial Theory. Series A 13 (3): 297–305. doi:10.1016/0097-3165(72)90063-5.

• Ketonen, Jussi (1978). “The structure of countable Boolean algebras”. Annals of Mathematics 108 (1): 41–89.doi:10.2307/1970929. JSTOR 1970929.

• Milner, E. C. (1985). “Basic WQO- and BQO-theory”. In Rival, I. Graphs and Order. The Role of Graphs inthe Theory of Ordered Sets and Its Applications. D. Reidel Publishing Co. pp. 487–502. ISBN 90-277-1943-8.

• Gallier, Jean H. (1991). “What’s so special about Kruskal’s theorem and the ordinal Γo? A survey of some re-sults in proof theory”. Annals of Pure and Applied Logic 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E.

28.9 See also• Better-quasi-ordering

• Prewellordering

• Well-order

28.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 85

28.10 Text and image sources, contributors, and licenses

28.10.1 Text• Process calculus Source: https://en.wikipedia.org/wiki/Process_calculus?oldid=653588725Contributors: Michael Hardy, AlexR, Theresa

knott, Charles Matthews, Zoicon5, Phil Boswell, Wmahan, Neilc, Solitude, Leibniz, Linas, Ruud Koot, MarSch, XP1, Jameshfisher,Vonkje, GangofOne, Wavelength, Koffieyahoo, CarlHewitt, Gareth Jones, RabidDeity, Jpbowen, SockPuppetVandal, Voidxor, Misza13,SmackBot, Chris the speller, Nbarth, Allan McInnes, Ezrakilty, Sam Staton, Blaisorblade, Thijs!bot, Dougher, Skraedingie, Barkjon,Roxy the dog, MystBot, Addbot, Tassedethe, Lightbot, Yobot, AnomieBOT, GrouchoBot, BehnazCh, Vasywriter, Iæfai, WikitanvirBot,Clayrat, Serketan, Bethnim, Helpful Pixie Bot, Ulidtko and Anonymous: 39

• Quasitransitive relation Source: https://en.wikipedia.org/wiki/Quasitransitive_relation?oldid=625289154Contributors: Patrick, CharlesMatthews, Giftlite, Btyner, Salix alba, SmackBot, Zahid Abdassabur, CRGreathouse, CBM, Gregbard, Sam Staton, Erik9bot, Deltahe-dron, Jochen Burghardt and Anonymous: 1

• Quotient by an equivalence relation Source: https://en.wikipedia.org/wiki/Quotient_by_an_equivalence_relation?oldid=657098718Contributors: Michael Hardy, TakuyaMurata, Bearcat, D.Lazard, K9re11, Iaritmioawp and Fimatic

• Rational consequence relation Source: https://en.wikipedia.org/wiki/Rational_consequence_relation?oldid=610309178 Contributors:Michael Hardy, Oleg Alexandrov, Reetep, SmackBot, CmdrObot, Gregbard, Enlil2, DavidCBryant, Carriearchdale and Tijfo098

• Reduct Source: https://en.wikipedia.org/wiki/Reduct?oldid=618205741 Contributors: John Baez, Spring Rubber, SmackBot, VaughanPratt, Tikiwont, Hans Adler, Backslash Forwardslash, Gf uip, EmausBot, Bgeron, Monkbot and Anonymous: 3

• Reflexive closure Source: https://en.wikipedia.org/wiki/Reflexive_closure?oldid=627004515 Contributors: Timwi, Arthur Rubin, Girl-withglasses, Henry Delforn (old), Addbot, Dawynn, Pcap, Renato sr, GrouchoBot, EmausBot, Jochen Burghardt, Vaizar and Anonymous:2

• Reflexive relation Source: https://en.wikipedia.org/wiki/Reflexive_relation?oldid=675919688Contributors: AxelBoldt, DavidSJ, Patrick,Wshun, TakuyaMurata, Looxix~enwiki, William M. Connolley, Charles Matthews, Josh Cherry, MathMartin, Henrygb, Tobias Berge-mann, Giftlite, Jason Quinn, Gubbubu, Urhixidur, Ascánder, Paul August, BenjBot, Spayrard, Spoon!, Jet57, LavosBacons, Wtmitchell,Bookandcoffee, Oleg Alexandrov, Joriki, Mel Etitis, LOL, MFH, Isnow, Audiovideo, Margosbot~enwiki, Fresheneesz, Chobot, Yurik-Bot, Laurentius, Maelin, Mathlaura, KarlHeg, Arthur Rubin, Isaac Dupree, Jdthood, Mhym, Ceosion, Mike Fikes, Fjbex, CRGreathouse,CBM, Gregbard, Farzaneh, Wikid77, JAnDbot, Policron, Joshua Issac, VolkovBot, Jackfork, Jamelan, Ocsenave, SieBot, Davidellerman,Henry Delforn (old), Hello71, Cuyaken, ClueBot, Ywanne, Da rulz07, SoxBot III, WikHead, Addbot, Download, Favonian, Luckas-bot, Yobot, Renato sr, Pkukiss, Galoubet, ArthurBot, Xqbot, Z0973, I dream of horses, RedBot, MastiBot, Gamewizard71, EmausBot,Dmayank, DimitriC, ClueBot NG, Kasirbot, Joel B. Lewis, BG19bot, Solomon7968, ChrisGualtieri, Eptified, Lerutit, Jochen Burghardt,Seanhalle and Anonymous: 37

• Relation algebra Source: https://en.wikipedia.org/wiki/Relation_algebra?oldid=657291951Contributors: Zundark, Michael Hardy, AugPi,Charles Matthews, Tobias Bergemann, Lethe, Mboverload, D6, Elwikipedista~enwiki, Giraffedata, AshtonBenson, Woohookitty, PaulCarpenter, BD2412, Rjwilmsi, Koavf, Tillmo, Ott2, Mhss, Concerned cynic, Nbarth, Jon Awbrey, Lambiam, Physis, Mets501, VaughanPratt, CBM, Gregbard, Sam Staton, King Bee, JustAGal, Balder ten Cate, David Eppstein, R'n'B, Leyo, Ramsey2006, Plasticup, John-Blackburne, The Tetrast, Linelor, Hans Adler, Addbot, QuadrivialMind, Yobot, AnomieBOT, Nastor, LilHelpa, Xqbot, Samppi111,Charvest, FrescoBot, Irmy, Sjcjoosten, SchreyP, Brad7777, Khazar2, Lerutit, RPI, JaconaFrere, SaltHerring, Some1Redirects4You andAnonymous: 35

• Relation construction Source: https://en.wikipedia.org/wiki/Relation_construction?oldid=564892457 Contributors: Charles Matthews,Carlossuarez46, Kaldari, Paul August, El C, DoubleBlue, Nihiltres, TeaDrinker, Gwernol, Wknight94, Closedmouth, SmackBot, JonAwbrey, JzG, Coredesat, Slakr, CBM, Gogo Dodo, Hut 8.5, Brusegadi, David Eppstein, Brigit Zilwaukee, Yolanda Zilwaukee, Ars Tottle,The Proposition That, Spellcast, Corvus cornix, Seb26, Maelgwnbot, Blanchardb, RABBU, REBBU, DEBBU, DABBUØ, Wolf of theSteppes, REBBUØ, Doubtentry, Education Is The Basis Of Law And Order, Bare In Mind, Preveiling Opinion Of Dominant OpinionGroup, VOCØ, Buchanan’s Navy Sec, Kaiba, Marsboat, Trainshift, Pluto Car, Unco Guid, Viva La Information Revolution!, FlowerMound Belle, Editortothemasses, Navy Pierre, Mrs. Lovett’s Meat Puppets, Unknown Justin, West Goshen Guy, Southeast Penna Poppa,Delaware Valley Girl, Erik9bot and Lerutit

• Representation (mathematics) Source: https://en.wikipedia.org/wiki/Representation_(mathematics)?oldid=647042099 Contributors:Michael Hardy, Giftlite, El C, Linas, SixWingedSeraph, Rjwilmsi, MarSch, Reyk, Magioladitis, A3nm, David Eppstein, PaulTanenbaum,Geometry guy, SieBot, Addbot, Twri, Trappist the monk, Helpful Pixie Bot, BattyBot, Mrt3366 and Anonymous: 2

• Separoid Source: https://en.wikipedia.org/wiki/Separoid?oldid=588579933 Contributors: Michael Hardy, Salix alba, SmackBot, Mhym,Gregbard, Magioladitis, Qworty, Hans Adler and Strausz~enwiki

• Series-parallel partial order Source: https://en.wikipedia.org/wiki/Series-parallel_partial_order?oldid=629384776 Contributors: Za-slav, A3nm, David Eppstein, Trappist the monk, Helpful Pixie Bot, Deltahedron and Anonymous: 1

• Surjective function Source: https://en.wikipedia.org/wiki/Surjective_function?oldid=672639860Contributors: AxelBoldt, Tarquin, Amil-lar, XJaM, Toby Bartels, Michael Hardy, Wshun, Pit~enwiki, Karada, Александър, Glenn, Jeandré du Toit, Hashar, Hawthorn, CharlesMatthews, Dysprosia, David Shay, Ed g2s, Phil Boswell, Aleph4, Robbot, Fredrik, Tobias Bergemann, Giftlite, Lethe, Jason Quinn,Jorge Stolfi, Matt Crypto, Keeyu, Rheun, MarkSweep, AmarChandra, Tsemii, TheObtuseAngleOfDoom, Vivacissamamente, Rich Farm-brough, Quistnix, Paul August, Bender235, Nandhp, Kevin Lamoreau, Larry V, Obradovic Goran, Dallashan~enwiki, ABCD, Schapel,Oleg Alexandrov, Tbsmith, Mindmatrix, LOL, Rjwilmsi, MarSch, FlaBot, Chobot, Manscher, Algebraist, Angus Lepper, Ksnortum,Rick Norwood, Sbyrnes321, SmackBot, Rotemliss, Bluebot, Javalenok, TedE, Soapergem, Dreadstar, Saippuakauppias, MickPurcell,16@r, Inquisitus, CBM, MatthewMain, Gregbard, Marqueed, Sam Staton, Pjvpjv, Prolog, Salgueiro~enwiki, JAnDbot, JamesBWatson,JJ Harrison, Martynas Patasius, MartinBot, TechnoFaye, Malerin, Dubhe.sk, Theabsurd, UnicornTapestry, Eliuha gmail.com, AnonymousDissident, SieBot, SLMarcus, Paolo.dL, Peiresc~enwiki, Classicalecon, UKoch, Watchduck, Bender2k14, SchreiberBike, Neuralwarp,Petru Dimitriu, Matthieumarechal, Kal-El-Bot, Addbot, Download, PV=nRT, ,ماني Zorrobot, Jarble, Legobot, Luckas-bot, Yobot, Frag-gle81, II MusLiM HyBRiD II, Xqbot, TechBot, Shvahabi, Raffamaiden, Omnipaedista, Applebringer, Erik9bot, LucienBOT, Tbhotch,Xnn, Jowa fan, EmausBot, PrisonerOfIce, WikitanvirBot, GoingBatty, Sasuketiimer, Maschen, Mjbmrbot, Anita5192, ClueBot NG,Helpful Pixie Bot, BG19bot, Cispyre, Lfahlberg, JPaestpreornJeolhlna, TranquilHope and Anonymous: 87

86 CHAPTER 28. WELL-QUASI-ORDERING

• Symmetric closure Source: https://en.wikipedia.org/wiki/Symmetric_closure?oldid=627004592 Contributors: Michael Hardy, Timwi,Addbot, Pcap, ZéroBot, YFdyh-bot, Jochen Burghardt and Anonymous: 2

• Symmetric relation Source: https://en.wikipedia.org/wiki/Symmetric_relation?oldid=676529262 Contributors: Patrick, Looxix~enwiki,William M. Connolley, Charles Matthews, MathMartin, Tobias Bergemann, Giftlite, Elektron, Ascánder, Paul August, Syp, Joriki, Is-now, Salix alba, Margosbot~enwiki, Fresheneesz, Roboto de Ajvol, Laurentius, Bota47, Arthur Rubin, Incnis Mrsi, Unyoyega, Jdthood,Vina-iwbot~enwiki, Gregbard, Thijs!bot, Mouchoir le Souris, David Eppstein, Jamelan, Henry Delforn (old), ClueBot, Libcub, Addbot,Luckas-bot, ArthurBot, Xqbot, Adavis444, RedBot, EmausBot, ZéroBot, Zap Rowsdower, EdoBot, DASHBotAV, 28bot, Kasirbot, No-suchforever, Aryan5496 and Anonymous: 18

• Ternary equivalence relation Source: https://en.wikipedia.org/wiki/Ternary_equivalence_relation?oldid=672070603Contributors: Mel-choir

• Ternary relation Source: https://en.wikipedia.org/wiki/Ternary_relation?oldid=677556030Contributors: Michael Hardy, Charles Matthews,Tobias Bergemann, Ancheta Wis, Abdull, Paul August, El C, Rgdboer, Versageek, Oleg Alexandrov, Jeffrey O. Gustafson, RxS, Dou-bleBlue, TeaDrinker, Wknight94, Closedmouth, Luk, Sardanaphalus, SmackBot, KnowledgeOfSelf, Melchoir, C.Fred, BiT, Aksi great,Nbarth, Jon Awbrey, Lambiam, JzG, Tim Q. Wells, Slakr, Politepunk, General Eisenhower, Happy-melon, Tawkerbot2, CBM, GogoDodo, Alaibot, Luna Santin, Hut 8.5, Transcendence, Brusegadi, David Eppstein, JoergenB, Santiago Saint James, Brigit Zilwaukee,Yolanda Zilwaukee, Fallopius Manque, Mike V, CardinalDan, Rei-bot, Seb26, GlobeGores, Lucien Odette, REBBU, RABBUØ, Wolfof the Steppes, REBBUØ, Doubtentry, DEBBUØ, Education Is The Basis Of Law And Order, Bare In Mind, Preveiling Opinion OfDominant Opinion Group, Hans Adler, Buchanan’s Navy Sec, Mr. Peabody’s Boy, Overstay, Marsboat, Unco Guid, Viva La InformationRevolution!, Autocratic Uzbek, Poke Salat Annie, Flower Mound Belle, Navy Pierre, Mrs. Lovett’s Meat Puppets, Unknown Justin, IPPhreely, West Goshen Guy, Delaware Valley Girl, Addbot, Jarble, Yobot, Vini 17bot5, AnomieBOT, In digma, Erik9bot, AManWithNo-Plan and Anonymous: 5

• Tolerance relation Source: https://en.wikipedia.org/wiki/Tolerance_relation?oldid=662333091 Contributors: Michael Hardy, Rjwilmsi,SmackBot, NickPenguin, Floridi~enwiki, Classicalecon, Gabrno, Clone200 and Anonymous: 1

• Total order Source: https://en.wikipedia.org/wiki/Total_order?oldid=673384381 Contributors: Damian Yerrick, AxelBoldt, Zundark,XJaM, Fritzlein, Patrick, Michael Hardy, Dori, AugPi, Dysprosia, Jitse Niesen, Greenrd, Zoicon5, Hyacinth, VeryVerily, Fibonacci,McKay, Aleph4, Gandalf61, MathMartin, Rursus, Tobias Bergemann, Giftlite, Mshonle~enwiki, Markus Krötzsch, Lethe, Waltpohl,DefLog~enwiki, Alberto da Calvairate~enwiki, Quarl, Elroch, Paul August, Susvolans, Army1987, Func, Cmdrjameson, Msh210, Pion,Joriki, MattGiuca, Yurik, OneWeirdDude, Salix alba, VKokielov, Mathbot, Margosbot~enwiki, Wastingmytime, Chobot, YurikBot,Hede2000, Tetracube, Rdore, Melchoir, Gelingvistoj, Mhss, Chris the speller, Bazonka, Jdthood, Javalenok, Michael Kinyon, Loadmas-ter, Mets501, JRSpriggs, George100, CRGreathouse, CBM, Thomasmeeks, Oryanw~enwiki, VectorPosse, JAnDbot, David Eppstein,Infovarius, Osquar F, PaulTanenbaum, SieBot, Ceroklis, Anchor Link Bot, Heinzi.at, WurmWoode, Universityuser, Palnot, Marc vanLeeuwen, Addbot, Tanhabot, AsphyxiateDrake, Luckas-bot, Yobot, Charlatino, White gecko, 1exec1, Infvwl, GrouchoBot, Jsjunkie,Quondum, D.Lazard, SporkBot, CocuBot, BG19bot, YumOooze, YFdyh-bot, Austinfeller, Mark viking, नितीश् चन्द्र and Anonymous:49

• Total relation Source: https://en.wikipedia.org/wiki/Total_relation?oldid=608974105 Contributors: Patrick, Charles Matthews, Dcoet-zee, Jitse Niesen, Tobias Bergemann, Lethe, Alberto da Calvairate~enwiki, Paul August, Ntmatter, Oleg Alexandrov, Joriki, Salix alba,Nneonneo, Mathbot, Fresheneesz, Bota47, Jdthood, Stotr~enwiki, JAnDbot, TXiKiBoT, Jamelan, Hans Adler, Erodium, Addbot, Fres-coBot, SporkBot, Helpful Pixie Bot, Deltahedron, Kephir and Anonymous: 11

• Transitive closure Source: https://en.wikipedia.org/wiki/Transitive_closure?oldid=675927373 Contributors: Awaterl, Vkuncak, Patrick,Michael Hardy, Charles Matthews, Timwi, Dcoetzee, Populus, Borislav, Tobias Bergemann, Giftlite, Fropuff, Matt Crypto, Alexf,Quickwik, Creidieki, Obradovic Goran, Oleg Alexandrov, Joriki, Neonfreon, Salix alba, AL SAM, Bgwhite, Ott2, Arthur Rubin, Plas-ticphilosopher, KnightRider~enwiki, Mhss, Plustgarten, Dreadstar, NeilFraser, Lyonsam, Loadmaster, JRSpriggs, CRGreathouse, CBM,ShelfSkewed, Gregbard, Girlwithglasses, Kirtag Hratiba, Thijs!bot, JAnDbot, A3nm, David Eppstein, Yavoh, Cometstyles, VolkovBot,Sdrucker, PaulTanenbaum, Jamelan, Tomaxer, LungZeno, Henry Delforn (old), DuaneLAnderson, CBM2, Classicalecon, ClueBot, Ben-der2k14, PixelBot, AmirOnWiki, MountainGoat8, Tayste, Addbot, Luckas-bot, AnomieBOT, BenzolBot, RedBot, MastiBot, Trappist themonk, Wizeguytristram, Quondum, Tijfo098, ClueBot NG, BG19bot, Solomon7968, Danwizard208, Dmitri L. Slabk., Vpieterse~enwiki,Seahen, Artdadamo and Anonymous: 26

• Transitive relation Source: https://en.wikipedia.org/wiki/Transitive_relation?oldid=677849508Contributors: Zundark, Patrick, MichaelHardy, Rp, Looxix~enwiki, Andres, Charles Matthews, Dcoetzee, Jitse Niesen, Fredrik, MathMartin, Tobias Bergemann, Giftlite, Ben-FrantzDale, Gubbubu, Chowbok, Paul August, MyNameIsNotBob, Spoon!, Polluks, DanShearer, Woohookitty, Linas, LOL, Isnow,Palica, Jérémie Lumbroso~enwiki, Salix alba, Mathbot, Fresheneesz, YurikBot, Laurentius, Sasuke Sarutobi, 48v, Bota47, Arthur Rubin,MaratL, Wasseralm, JJL, SmackBot, InverseHypercube, Nbarth, Wen D House, Cybercobra, Jóna Þórunn, Lambiam, Coredesat, Lyon-sam, Cbuckley, CRGreathouse, Aggarwal kshitij, CBM, Thomasmeeks, Gogo Dodo, Tawkerbot4, AntiVandalBot, Mhaitham.shammaa,MER-C, .anacondabot, Magioladitis, Albmont, David Eppstein, Edward321, MartinBot, Extransit, Tomaz.slivnik, Policron, VolkovBot,AThomas203, Jamelan, Cnilep, SieBot, Paradoctor, Henry Delforn (old), Anchor Link Bot, ClueBot, Tomvanderweide, Sarbogard, Ot-tawahitech, Alexbot, Wikibojopayne, Pa68, SilvonenBot, Addbot, Luckas-bot, Yobot, Ptbotgourou, Materialscientist, GrouchoBot, Und-soweiter, RedBot, Katovatzschyn, EmausBot, Slightsmile, IGeMiNix, ChuispastonBot, ClueBot NG, Pars99, Sourabh.khot, Justincheng12345-bot, Lerutit, Loraof and Anonymous: 61

• Trichotomy (mathematics) Source: https://en.wikipedia.org/wiki/Trichotomy_(mathematics)?oldid=646849767Contributors: Zundark,Patrick, Michael Hardy, Arthur Frayn, Casu Marzu, Henrygb, UtherSRG, Tobias Bergemann, Macrakis, Paul August, Oleg Alexandrov,VKokielov, Michael Slone, Bota47, JJL, SmackBot, Mhss, Colonies Chris, Jdthood, Cybercobra, Courcelles, JRSpriggs, Meng.benjamin,Gregbard, Difluoroethene, David Cherney, AntiVandalBot, Indeed123, Minnnnng, YohanN7, MilesAgain, Addbot, Luckas-bot, AnomieBOT,Götz, Xqbot, Constructive editor, ComputScientist, Þjóðólfr, Tkuvho, SporkBot, Paulmiko, Helpful Pixie Bot, Zorglub x and Anonymous:23

• Unimodality Source: https://en.wikipedia.org/wiki/Unimodality?oldid=670019894Contributors: Michael Hardy, Komap, Henrygb, Com-mander Keane, Rjwilmsi, Bhny, Tropylium, Wen D House, Henning Makholm, JzG, CRGreathouse, Magioladitis, David Eppstein, DrMi-cro, Asaduzaman, Melcombe, Muhandes, Juanm55, Addbot, Nate Wessel, Yobot, Flavonoid, Isheden, FrescoBot, Ofir michael, Duoduo-duo, Gypped, Tgoossens, Helpful Pixie Bot, BG19bot, Op47, CitationCleanerBot, BattyBot, ChrisGualtieri, Iamyatin and Anonymous:14

28.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 87

• Weak ordering Source: https://en.wikipedia.org/wiki/Weak_ordering?oldid=640088882 Contributors: Patrick, Michael Hardy, Chinju,Dcoetzee, MathMartin, Tobias Bergemann, Pretzelpaws, AlphaEtaPi, Zaslav, Aisaac, YurikBot, Gadget850, Modify, Oli Filth, Jdthood,Chlewbot, Radiant chains, Jafet, CRGreathouse, Sdorrance, Gregbard, Widefox, Medinoc, Zeitlupe, David Eppstein, Jonathanrcoxhead,Watchduck, Addbot, Kne1p, Forich, Citation bot, ArthurBot, Howard McCay, Citation bot 1, SporkBot, Joel B. Lewis, Helpful Pixie Bot,Jochen Burghardt, JustBerry and Anonymous: 12

• Well-founded relation Source: https://en.wikipedia.org/wiki/Well-founded_relation?oldid=659840399 Contributors: The Anome, Ap,Michael Hardy, Dominus, TakuyaMurata, Cyp, Charles Matthews, VeryVerily, Aleph4, Mountain, Tobias Bergemann, Filemon, Marekpetrik,Lethe, Lupin, Waltpohl, Mani1, Paul August, EmilJ, Nahabedere, MZMcBride, R.e.b., FlaBot, Trovatore, Mikeblas, Crasshopper, Mar-lasdad, That Guy, From That Show!, SmackBot, Ron.garcia, Nbarth, Mmehdi.g, Mets501, CBM, WillowW, Pgagge, JAnDbot, Albmont,Leyo, Reedy Bot, Alexsmail, Thehotelambush, Mutilin, Addbot, KamikazeBot, RibotBOT, FrescoBot, Involutive-revolution, MerlIwBot,Wohlfundi, YiFeiBot, Some1Redirects4You and Anonymous: 22

• Well-order Source: https://en.wikipedia.org/wiki/Well-order?oldid=672639066 Contributors: AxelBoldt, Mav, Zundark, Josh Grosse,Patrick, Michael Hardy, David Martland, Dominus, TakuyaMurata, Andres, Vargenau, Revolver, Charles Matthews, Timwi, Populus,Aleph4, R3m0t, MathMartin, Tobias Bergemann, Tosha, Giftlite, Dbenbenn, Ian Maxwell, Lethe, Arturus~enwiki, Jorend, Karl-Henner,Rich Farmbrough, Luqui, Paul August, Sligocki, RJFJR, Eyu100, Salix alba, FlaBot, Margosbot~enwiki, Chobot, YurikBot, Hairy Dude,Trovatore, Obey, Bota47, Arthur Rubin, MullerHolk, Ghazer~enwiki, GrinBot~enwiki, KnightRider~enwiki, Alan McBeth, Gelingvistoj,Mhss, Nbarth, Loodog, Jim.belk, Loadmaster, JRSpriggs, CBM, WeggeBot, Myasuda, Thijs!bot, Nadav1, Escarbot, Albmont, Odexios,VolkovBot, Don4of4, SieBot, Rumping, Fyyer, Bender2k14, His Wikiness, Palnot, Addbot, Luckas-bot, Yobot, Ptbotgourou, Jarmiz,Xqbot, GrouchoBot, Miyagawa, Adrionwells, RjwilmsiBot, Honestrosewater, Hunterbd, SporkBot, Misshamid, ChrisGualtieri, Khazar2,Jose Brox and Anonymous: 29

• Well-quasi-ordering Source: https://en.wikipedia.org/wiki/Well-quasi-ordering?oldid=677173775Contributors: Patrick, Chinju, CharlesMatthews, Tobias Bergemann, Peter Kwok, Rich Farmbrough, Paul August, EmilJ, R.e.b., Open2universe, PhS, That Guy, From ThatShow!, Mets501, Pierre de Lyon, David Eppstein, Kope, R'n'B, Alexwright, Fcarreiro, Niceguyedc, Palnot, Addbot, DOI bot, Citationbot, FrescoBot, Citation bot 1, Gongfarmerzed, John of Reading, ZéroBot, Ɯ, Mastergreg82, Paolo Lipparini, CitationCleanerBot, JeffErickson, Mark viking, Anrnusna, ,תמשל Gasarch and Anonymous: 5

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88 CHAPTER 28. WELL-QUASI-ORDERING

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