Demonic compositionFrom Wikipedia, the free encyclopedia

Contents

1 Asymmetric relation 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Bijection 32.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Binary relation 93.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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3.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Composition of relations 194.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Join: another form of composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Demonic composition 215.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Inverse relation 226.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6.1.1 Inverse relation of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7 Partially ordered set 247.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 267.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.10 Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.11 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.12 Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.13 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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7.14 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.15 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.17 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Reflexive relation 318.1 Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.3 Number of reflexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.4 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

9 Surjective function 359.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9.3.1 Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.3.4 Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

10 Total order 4110.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

10.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

10.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 4410.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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10.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

11 Total relation 4611.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.2 Properties and related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

12 Transitive relation 4812.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

12.3.1 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3.3 Properties that require transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

12.4 Counting transitive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

12.6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

12.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

13 Weak ordering 5113.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.2 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

13.2.1 Strict weak orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.2.2 Total preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5313.2.3 Ordered partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5313.2.4 Representation by functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

13.3 Related types of ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.4 All weak orders on a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

13.4.1 Combinatorial enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.4.2 Adjacency structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

13.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5613.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5613.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 57

13.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5713.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5813.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 1

Asymmetric relation

In mathematics an asymmetric relation is a binary relation on a set X where:

• For all a and b in X, if a is related to b, then b is not related to a.[1]

In mathematical notation, this is:

∀a, b ∈ X, aRb ⇒ ¬(bRa)

1.1 Examples

An example is < (less-than): if x < y, then necessarily y is not less than x. In fact, one of Tarski’s axioms characterizingthe real numbers R is that < over R is asymmetric.An asymmetric relation need not be total. For example, strict subset or ⊊ is asymmetric, and neither of the sets {1,2}and {3,4} is a strict subset of the other. In general, every strict partial order is asymmetric, and conversely, everytransitive asymmetric relation is a strict partial order.Not all asymmetric relations are strict partial orders, however. An example of an asymmetric intransitive relation isthe rock-paper-scissors relation: if X beats Y, then Y does not beat X, but no one choice wins all the time.The ≤ (less than or equal) operator, on the other hand, is not asymmetric, because reversing x ≤ x produces x ≤ xand both are true. In general, any relation in which x R x holds for some x (that is, which is not irreflexive) is also notasymmetric.Asymmetric is not the same thing as “not symmetric": a relation can be neither symmetric nor asymmetric, such as≤, or can be both, only in the case of the empty relation (vacuously).

1.2 Properties• A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[2]

• Restrictions and inverses of asymmetric relations are also asymmetric. For example, the restriction of < fromthe reals to the integers is still asymmetric, and the inverse > of < is also asymmetric.

• A transitive relation is asymmetric if and only if it is irreflexive:[3] if a R b and b R a, transitivity gives a R a,contradicting irreflexivity.

1.3 See also• Symmetric relation

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2 CHAPTER 1. ASYMMETRIC RELATION

• Antisymmetric relation

• Symmetry

• Symmetry in mathematics

1.4 References[1] Gries, David; Schneider, Fred B. (1993), A Logical Approach to Discrete Math, Springer-Verlag, p. 273.

[2] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[3] 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”.

Chapter 2

Bijection

X 1

2

3

4

YD

B

C

A

A bijective function, f: X→ Y, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D.

In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elementsof two sets, where every element of one set is paired with exactly one element of the other set, and every elementof the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical

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4 CHAPTER 2. BIJECTION

terms, a bijective function f: X→ Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.A bijection from the setX to the set Y has an inverse function from Y toX. IfX and Y are finite sets, then the existenceof a bijection means they have the same number of elements. For infinite sets the picture is more complicated, leadingto the concept of cardinal number, a way to distinguish the various sizes of infinite sets.A bijective function from a set to itself is also called a permutation.Bijective functions are essential tomany areas ofmathematics including the definitions of isomorphism, homeomorphism,diffeomorphism, permutation group, and projective map.

2.1 Definition

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

For a pairing between X and Y (where Y need not be different from X) to be a bijection, four properties must hold:

1. each element of X must be paired with at least one element of Y,

2. no element of X may be paired with more than one element of Y,

3. each element of Y must be paired with at least one element of X, and

4. no element of Y may be paired with more than one element of X.

Satisfying properties (1) and (2) means that a bijection is a function with domain X. It is more common to seeproperties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y.Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions).Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injectivefunctions).[1] With this terminology, a bijection is a function which is both a surjection and an injection, or usingother words, a bijection is a function which is both “one-to-one” and “onto”.

2.2 Examples

2.2.1 Batting line-up of a baseball team

Consider the batting line-up of a baseball team (or any list of all the players of any sports team). The set X will be thenine players on the team and the set Y will be the nine positions in the batting order (1st, 2nd, 3rd, etc.) The “pairing”is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere inthe list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says thatfor each position in the order, there is some player batting in that position and property (4) states that two or moreplayers are never batting in the same position in the list.

2.2.2 Seats and students of a classroom

In a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks themall to be seated. After a quick look around the room, the instructor declares that there is a bijection between the setof students and the set of seats, where each student is paired with the seat they are sitting in. What the instructorobserved in order to reach this conclusion was that:

1. Every student was in a seat (there was no one standing),

2. No student was in more than one seat,

3. Every seat had someone sitting there (there were no empty seats), and

4. No seat had more than one student in it.

2.3. MORE MATHEMATICAL EXAMPLES AND SOME NON-EXAMPLES 5

The instructor was able to conclude that there were just as many seats as there were students, without having to counteither set.

2.3 More mathematical examples and some non-examples• For any set X, the identity function 1X: X→ X, 1X(x) = x, is bijective.

• The function f: R→ R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x)= y. In more generality, any linear function over the reals, f: R→ R, f(x) = ax + b (where a is non-zero) is abijection. Each real number y is obtained from (paired with) the real number x = (y - b)/a.

• The function f: R → (-π/2, π/2), given by f(x) = arctan(x) is bijective since each real number x is pairedwith exactly one angle y in the interval (-π/2, π/2) so that tan(y) = x (that is, y = arctan(x)). If the codomain(-π/2, π/2) was made larger to include an integer multiple of π/2 then this function would no longer be onto(surjective) since there is no real number which could be paired with the multiple of π/2 by this arctan function.

• The exponential function, g: R→ R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) =−1, showing that g is not onto (surjective). However if the codomain is restricted to the positive real numbersR+ ≡ (0,+∞) , then g becomes bijective; its inverse (see below) is the natural logarithm function ln.

• The function h: R→ R+, h(x) = x2 is not bijective: for instance, h(−1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to R+

0 ≡ [0,+∞) , then h becomes bijective; its inverseis the positive square root function.

2.4 Inverses

A bijection f with domainX (“functionally” indicated by f: X→Y) also defines a relation starting in Y and going toX(by turning the arrows around). The process of “turning the arrows around” for an arbitrary function does not usuallyyield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y.Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inversefunction exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function isinvertible if and only if it is a bijection.Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition

for every y in Y there is a unique x in X with y = f(x).

Continuing with the baseball batting line-up example, the function that is being defined takes as input the name ofone of the players and outputs the position of that player in the batting order. Since this function is a bijection, it hasan inverse function which takes as input a position in the batting order and outputs the player who will be batting inthat position.

2.5 Composition

The composition g ◦ f of two bijections f: X → Y and g: Y → Z is a bijection. The inverse of g ◦ f is (g ◦ f)−1 =

(f−1) ◦ (g−1) .Conversely, if the composition g ◦ f of two functions is bijective, we can only say that f is injective and g is surjective.

2.6 Bijections and cardinality

If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y havethe same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of “same number ofelements” (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number,a way to distinguish the various sizes of infinite sets.

6 CHAPTER 2. BIJECTION

X1

2

3

YD

B

C

A

ZP

Q

R

A bijection composed of an injection (left) and a surjection (right).

2.7 Properties

• A function f: R→ R is bijective if and only if its graph meets every horizontal and vertical line exactly once.

• If X is a set, then the bijective functions from X to itself, together with the operation of functional composition(∘), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).

• Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of thecodomain with cardinality |B|, one has the following equalities:

|f(A)| = |A| and |f−1(B)| = |B|.

• If X and Y are finite sets with the same cardinality, and f: X → Y, then the following are equivalent:

1. f is a bijection.2. f is a surjection.3. f is an injection.

• For a finite set S, there is a bijection between the set of possible total orderings of the elements and the set ofbijections from S to S. That is to say, the number of permutations of elements of S is the same as the numberof total orderings of that set—namely, n!.

2.8 Bijections and category theory

Bijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are notalways the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphismsmust be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphismswhich are bijective homomorphisms.

2.9. GENERALIZATION TO PARTIAL FUNCTIONS 7

2.9 Generalization to partial functions

The notion of one-one correspondence generalizes to partial functions, where they are called partial bijections,although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partialfunction is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverseto be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set iscalled the symmetric inverse semigroup.[2]

Another way of defining the same notion is to say that a partial bijection from A to B is any relation R (which turnsout to be a partial function) with the property that R is the graph of a bijection f:A′→B′, where A′ is a subset of Aand likewise B′⊆B.[3]

When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[4] Anexample is theMöbius transformation simply defined on the complex plane, rather than its completion to the extendedcomplex plane.[5]

2.10 Contrast withThis list is incomplete; you can help by expanding it.

• Multivalued function

2.11 See also• Injective function

• Surjective function

• Bijection, injection and surjection

• Symmetric group

• Bijective numeration

• Bijective proof

• Cardinality

• Category theory

• Ax–Grothendieck theorem

2.12 Notes[1] There are names associated to properties (1) and (2) as well. A relation which satisfies property (1) is called a total relation

and a relation satisfying (2) is a single valued relation.

[2] Christopher Hollings (16 July 2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups.American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[3] Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge UniversityPress. p. 289. ISBN 978-0-521-44179-7.

[4] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.

[5] John Meakin (2007). “Groups and semigroups: connections and contrasts”. In C.M. Campbell, M.R. Quick, E.F. Robert-son, G.C. Smith. Groups St Andrews 2005 Volume 2. Cambridge University Press. p. 367. ISBN 978-0-521-69470-4.preprint citing Lawson,M.V. (1998). “TheMöbius InverseMonoid”. Journal of Algebra 200 (2): 428. doi:10.1006/jabr.1997.7242.

8 CHAPTER 2. BIJECTION

2.13 References

This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory.Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic maybe found in any of these:

• Wolf (1998). Proof, Logic and Conjecture: A Mathematician’s Toolbox. Freeman.

• Sundstrom (2003). Mathematical Reasoning: Writing and Proof. Prentice-Hall.

• Smith; Eggen; St.Andre (2006). A Transition to Advanced Mathematics (6th Ed.). Thomson (Brooks/Cole).

• Schumacher (1996). Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley.

• O'Leary (2003). The Structure of Proof: With Logic and Set Theory. Prentice-Hall.

• Morash. Bridge to Abstract Mathematics. Random House.

• Maddox (2002). Mathematical Thinking and Writing. Harcourt/ Academic Press.

• Lay (2001). Analysis with an introduction to proof. Prentice Hall.

• Gilbert; Vanstone (2005). An Introduction to Mathematical Thinking. Pearson Prentice-Hall.

• Fletcher; Patty. Foundations of Higher Mathematics. PWS-Kent.

• Iglewicz; Stoyle. An Introduction to Mathematical Reasoning. MacMillan.

• Devlin, Keith (2004). Sets, Functions, and Logic: An Introduction to Abstract Mathematics. Chapman & Hall/CRC Press.

• D'Angelo; West (2000). Mathematical Thinking: Problem Solving and Proofs. Prentice Hall.

• Cupillari. The Nuts and Bolts of Proofs. Wadsworth.

• Bond. Introduction to Abstract Mathematics. Brooks/Cole.

• Barnier; Feldman (2000). Introduction to Advanced Mathematics. Prentice Hall.

• Ash. A Primer of Abstract Mathematics. MAA.

2.14 External links• Hazewinkel, Michiel, ed. (2001), “Bijection”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Weisstein, Eric W., “Bijection”, MathWorld.

• Earliest Uses of Some of theWords ofMathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.

Chapter 3

Binary relation

“Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation § Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subsetof A × B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and“divides” in arithmetic, "is congruent to" in geometry, “is adjacent to” in graph theory, “is orthogonal to” in linearalgebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZ×Z×Z is “lies between ... and ...”, containing e.g. the triples (5,2,8), (5,8,2), and (−4,9,−7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in settheory, without running into logical inconsistencies such as Russell’s paradox.

3.1 Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X × Y for the set of pairs of G.The order of the elements in each pair ofG is important: if a ≠ b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X × Y, and “from X to Y" must always be either specified or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

9

10 CHAPTER 3. BINARY RELATION

3.1.1 Is a relation more than its graph?

According to the definition above, two relations with identical graphs but different domains or different codomainsare considered different. For example, ifG = {(1, 2), (1, 3), (2, 7)} , then (Z,Z, G) , (R,N, G) , and (N,R, G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least oney such that (x, y) ∈ R , the range of R is defined as the set of all y such that there exists at least one x such that(x, y) ∈ R , and the field of R is the union of its domain and its range.[2][3][4]

A special case of this difference in points of view applies to the notion of function. Many authors insist on distin-guishing between a function’s codomain and its range. Thus, a single “rule,” like mapping every real number x tox2, can lead to distinct functions f : R → R and f : R → R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the definitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodefinitions usually matters only in very formal contexts, like category theory.

3.1.2 Example

Example: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation “is owned by” is given as

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by ₐ RJₒ means that the ball is owned by John.Two different relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})

is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identified or even defined as G(R) and “an ordered pair (x, y) ∈ G(R)" is usually denoted as"(x, y) ∈ R".

3.2 Special types of binary relations

Some important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be different sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

• injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5and z = +5 to y = 25.

• functional (also called univalent[8] or right-unique[7] or right-definite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=−5 and z=+5.

3.2. SPECIAL TYPES OF BINARY RELATIONS 11

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

• one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

• left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is differentfrom the definition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = −14 to any real number y.

• surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:

12 CHAPTER 3. BINARY RELATION

• A function: a relation that is functional and left-total. Both the green and the red relation are functions.

• An injective function: a relation that is injective, functional, and left-total.

• A surjective function or surjection: a relation that is functional, left-total, and right-total.

• A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

3.2.1 Difunctional

Less commonly encountered is the notion of difunctional (or regular) relation, defined as a relation R such thatR=RR−1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element x∈X to a set xR = { y∈Y| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can define the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R ∩ x2R ≠ ∅ implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A → Cand g: B→ C and then define the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) ∈ A × B | f(a) = g(b) }. Every difunctional relation R ⊆ A × B arises as the joint kernel of two functionsf: A→ C and g: B→ C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]Other authors however use the term “rectangular” to denote any heterogeneous relation whatsoever.[6]

3.3 Relations over a set

If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely studied in graph theory, where they are known as simple directed graphs permitting loops.The set of all binary relations Rel(X) on a set X is the set 2X × X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

• reflexive: for all x in X it holds that xRx. For example, “greater than or equal to” (≥) is a reflexive relation but“greater than” (>) is not.

• irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but ≥ is not.

• coreflexive: for all x and y in X it holds that if xRy then x = y. An example of a coreflexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reflexive and coreflexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

• symmetric: for all x and y in X it holds that if xRy then yRx. “Is a blood relative of” is a symmetric relation,because x is a blood relative of y if and only if y is a blood relative of x.

3.4. OPERATIONS ON BINARY RELATIONS 13

• antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric (so is >, butonly because the condition in the definition is always false).[18]

• asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreflexive.[19] For example, > is asymmetric, but ≥ is not.

• transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. A transitive relation is irreflexive if andonly if it is asymmetric.[20] For example, “is ancestor of” is transitive, while “is parent of” is not.

• total: for all x and y in X it holds that xRy or yRx (or both). This definition for total is different from left totalin the previous section. For example, ≥ is a total relation.

• trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation “divides” on natural numbers is not.[21]

• Right Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz.

• Left Euclidean: for all x, y and z in X it holds that if yRx and zRx, then yRz.

• Euclidean: An Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because ifx=y and x=z, then y=z.

• serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the definition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

• set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily beingreflexive) is called a partial equivalence relation.A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

3.4 Operations on binary relations

If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y :

• Union: R ∪ S ⊆ X × Y, defined as R ∪ S = { (x, y) | (x, y) ∈ R or (x, y) ∈ S }. For example, ≥ is the union of >and =.

• Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = { (x, y) | (x, y) ∈ R and (x, y) ∈ S }.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relationover X and Z: (see main article composition of relations)

• Composition: S ∘ R, also denoted R ; S (or more ambiguously R ∘ S), defined as S ∘ R = { (x, z) | there existsy ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. The order of R and S in the notation S ∘ R, used here agrees withthe standard notational order for composition of functions. For example, the composition “is mother of” ∘ “isparent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “isgrandmother of”.

14 CHAPTER 3. BINARY RELATION

A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin ≥.If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

• Inverse or converse: R −1, defined as R −1 = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, “is less than” (<) is theinverse of “is greater than” (>).

If R is a binary relation over X, then each of the following is a binary relation over X:

• Reflexive closure: R =, defined as R = = { (x, x) | x ∈ X } ∪ R or the smallest reflexive relation over X containingR. This can be proven to be equal to the intersection of all reflexive relations containing R.

• Reflexive reduction: R ≠, defined as R ≠ = R \ { (x, x) | x ∈ X } or the largest irreflexive relation over Xcontained in R.

• Transitive closure: R +, defined as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

• Transitive reduction: R −, defined as a minimal relation having the same transitive closure as R.

• Reflexive transitive closure: R *, defined as R * = (R +) =, the smallest preorder containing R.

• Reflexive transitive symmetric closure: R ≡, defined as the smallest equivalence relation over X containingR.

3.4.1 Complement

If R is a binary relation over X and Y, then the following too:

• The complement S is defined as x S y if not x R y. For example, on real numbers, ≤ is the complement of >.

The complement of the inverse is the inverse of the complement.If X = Y, the complement has the following properties:

• If a relation is symmetric, the complement is too.

• The complement of a reflexive relation is irreflexive and vice versa.

• The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

3.4.2 Restriction

The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x andy are in S.If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of “is parent of” is “is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.

3.5. SETS VERSUS CLASSES 15

Also, the various concepts of completeness (not to be confused with being “total”) do not carry over to restrictions.For example, on the set of real numbers a property of the relation "≤" is that every non-empty subset S of R with anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to theset of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain(codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

3.4.3 Algebras, categories, and rewriting systems

Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finiteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

3.5 Sets versus classes

Certain mathematical “relations”, such as “equal to”, “member of”, and “subset of”, cannot be understood to be binaryrelations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of “equality” as a binary relation =, wemust take the domain and codomain to be the “class of all sets”, which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis to select a “large enough” set A, that contains all the objects of interest, and work with the restriction =A instead of=. Similarly, the “subset of” relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set ofa specific set A): the resulting set relation can be denoted ⊆A. Also, the “member of” relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown thatassuming ∈ to be defined on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory,and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership,and subset are binary relations without special comment. (A minor modification needs to be made to the concept ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this definition one can for instance define a functionrelation between every set and its power set.

3.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):Notes:

• The number of irreflexive relations is the same as that of reflexive relations.

• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.

• The number of strict weak orders is the same as that of total preorders.

• The total orders are the partial orders that are also total preorders. The number of preorders that are neithera partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

• the number of equivalence relations is the number of partitions, which is the Bell number.

16 CHAPTER 3. BINARY RELATION

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

3.7 Examples of common binary relations

• order relations, including strict orders:

• greater than• greater than or equal to• less than• less than or equal to• divides (evenly)• is a subset of

• equivalence relations:

• equality• is parallel to (for affine spaces)• is in bijection with• isomorphy

• dependency relation, a finite, symmetric, reflexive relation.

• independency relation, a symmetric, irreflexive relation which is the complement of some dependency relation.

3.8 See also

• Confluence (term rewriting)

• Hasse diagram

• Incidence structure

• Logic of relatives

• Order theory

• Triadic relation

3.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; PanosM. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science&BusinessMedia. pp. 299–300. ISBN 978-0-387-74758-3.

3.10. REFERENCES 17

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:

• Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

• Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN 978-0-13-460643-9.

• Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 21–22. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Mäs, Stephan (2007), “Reasoning on Spatial Semantic Integrity Constraints”, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285–302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of “correspondence” here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). “Semantics of Fuzzy Sets in Rough Set Theory”. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). “Coalgebraic Simulations and Congruences”. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[16] M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2

[19] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[20] 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). This source refers to asymmetric relations as “strictlyantisymmetric”.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). “Generalization of rough sets using relationships between attribute values” (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

3.10 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

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

18 CHAPTER 3. BINARY RELATION

3.11 External links• Hazewinkel, Michiel, ed. (2001), “Binary relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 4

Composition of relations

In mathematics, the composition of binary relations is a concept of forming a new relation S ∘ R from two givenrelations R and S, having as its most well-known special case the composition of functions.

4.1 Definition

If R ⊆ X × Y and S ⊆ Y × Z are two binary relations, then their composition S ◦R is the relation

S ◦R = {(x, z) ∈ X × Z | ∃y ∈ Y : (x, y) ∈ R ∧ (y, z) ∈ S}.

In other words, S ◦ R ⊆ X × Z is defined by the rule that says (x, z) ∈ S ◦ R if and only if there is an elementy ∈ Y such that xR y S z (i.e. (x, y) ∈ R and (y, z) ∈ S ).In particular fields, authors might denote by R ∘ S what is defined here to be S ∘ R. The convention chosen here issuch that function composition (with the usual notation) is obtained as a special case, when R and S are functionalrelations. Some authors[1] prefer to write ◦l and ◦r explicitly when necessary, depending whether the left or the rightrelation is the first one applied.A further variation encountered in computer science is the Z notation: ◦ is used to denote the traditional (right)composition, but ; (a fat open semicolon with Unicode code point U+2A3E) denotes left composition.[2][3] This useof semicolon coincides with the notation for function composition used (mostly by computer scientists) in Categorytheory,[4] as well as the notation for dynamic conjunction within linguistic dynamic semantics.[5] The semicolonnotation (with this semantic) was introduced by Ernst Schröder in 1895.[6]

The binary relations R ⊆ X × Y are sometimes regarded as the morphisms R : X → Y in a category Rel whichhas the sets as objects. In Rel, composition of morphisms is exactly composition of relations as defined above. Thecategory Set of sets is a subcategory of Rel that has the same objects but fewer morphisms. A generalization of thisis found in the theory of allegories.

4.2 Properties

Composition of relations is associative.The inverse relation of S ∘ R is (S ∘ R)−1 = R−1 ∘ S−1. This property makes the set of all binary relations on a set asemigroup with involution.The compose of (partial) functions (i.e. functional relations) is again a (partial) function.If R and S are injective, then S ∘ R is injective, which conversely implies only the injectivity of R.If R and S are surjective, then S ∘ R is surjective, which conversely implies only the surjectivity of S.The set of binary relations on a set X (i.e. relations from X to X) together with (left or right) relation compositionforms a monoid with zero, where the identity map on X is the neutral element, and the empty set is the zero element.

19

20 CHAPTER 4. COMPOSITION OF RELATIONS

4.3 Join: another form of composition

Main article: Join (relational algebra)

Other forms of composition of relations, which apply to general n-place relations instead of binary relations, arefound in the join operation of relational algebra. The usual composition of two binary relations as defined here canbe obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middlecomponent.

4.4 See also• Binary relation

• Relation algebra

• Demonic composition

• Function composition

• Join (SQL)

• Logical matrix

4.5 Notes[1] Kilp, Knauer & Mikhalev, p. 7

[2] ISO/IEC 13568:2002(E), p. 23

[3] http://www.fileformat.info/info/unicode/char/2a3e/index.htm

[4] http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf, p. 6

[5] http://plato.stanford.edu/entries/dynamic-semantics/#EncDynTypLog

[6] Paul Taylor (1999). Practical Foundations of Mathematics. Cambridge University Press. p. 24. ISBN 978-0-521-63107-5.A free HTML version of the book is available at http://www.cs.man.ac.uk/~{}pt/Practical_Foundations/

4.6 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

Chapter 5

Demonic composition

In mathematics, demonic composition is an operation on binary relations that is somewhat comparable to ordinarycomposition of relations but is robust to refinement of the relations into (partial) functions or injective relations.Unlike ordinary composition of relations, demonic composition is not associative.

5.1 Definition

SupposeR is a binary relation betweenX andY and S is a relation betweenY and Z. Their right demonic compositionR ;→ S is a relation between X and Z. Its graph is defined as

{(x, z) | x (S ◦R) z ∧ ∀y ∈ Y (xR y ⇒ y S z)}.

Conversely, their left demonic composition R ;← S is defined by

{(x, z) | x (S ◦R) z ∧ ∀y ∈ Y (y S z ⇒ xR y)}.

5.2 References• Backhouse, Roland; van der Woude, Jaap (1993), “Demonic operators and monotype factors”, MathematicalStructures in Computer Science 3 (4): 417–433, doi:10.1017/S096012950000030X, MR 1249420.

21

Chapter 6

Inverse relation

For inverse relationships in statistics, see negative relationship.

In mathematics, the inverse relation of a binary relation is the relation that occurs when the order of the elements isswitched in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms,ifX and Y are sets and L ⊆ X × Y is a relation from X to Y then L−1 is the relation defined so that y L−1 x if andonly if xLy . In set-builder notation, L−1 = {(y, x) ∈ Y ×X | (x, y) ∈ L} .The notation comes by analogy with that for an inverse function. Although many functions do not have an inverse;every relation does have a unique inverse. Despite the notation and terminology, the inverse relation is not an inversein the sense of group inverse; the unary operation that maps a relation to the inverse relation is however an involution,so it induces the structure of a semigroup with involution on the binary relations on a set, or more generally induces adagger category on the category of relations as detailed below. As a unary operation, taking the inverse (sometimescalled inversion) commutes however with the order-related operations of relation algebra, i.e. it commutes withunion, intersection, complement etc.The inverse relation is also called the converse relation or transpose relation— the latter in view of its similaritywith the transpose of a matrix.[1] It has also been called the opposite or dual of the original relation.[2] Other notationsfor the inverse relation include LC , LT , L~ or L̆ or L° or L∨.

6.1 Examples

For usual (maybe strict or partial) order relations, the converse is the naively expected “opposite” order, e.g. ≤−1=≥, <−1= > , etc.

6.1.1 Inverse relation of a function

A function is invertible if and only if its inverse relation is a function, in which case the inverse relation is the inversefunction.The inverse relation of a function f : X → Y is the relation f−1 : Y → X defined by graph f−1 = {(y, x) | y =f(x)} .This is not necessarily a function: One necessary condition is that f be injective, since else f−1 is multi-valued. Thiscondition is sufficient for f−1 being a partial function, and it is clear that f−1 then is a (total) function if and only iff is surjective. In that case, i.e. if f is bijective, f−1 may be called the inverse function of f.

6.2 Properties

In the monoid of binary endorelations on a set (with the binary operation on relations being the composition ofrelations), the inverse relation does not satisfy the definition of an inverse from group theory, i.e. if L is an arbitrary

22

6.3. SEE ALSO 23

relation on X, then L ◦L−1 does not equal the identity relation on X in general. The inverse relation does satisfy the(weaker) axioms of a semigroup with involution: (L−1)−1 = L and (L ◦R)−1 = R−1 ◦ L−1 .[3]

Since one may generally consider relations between different sets (which form a category rather than a monoid,namely the category of relationsRel), in this context the inverse relation conforms to the axioms of a dagger category(aka category with involution).[3] A relation equal to its inverse is a symmetric relation; in the language of daggercategories, it is self-adjoint.Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relationsas sets), and actually an involutive quantale. Similarly, the category of heterogenous relations, Rel is also an orderedcategory.[3]

In relation algebra (which is an abstraction of the properties of the algebra of endorelations on a set), inversion (theoperation of taking the inverse relation) commutes with other binary operations of union and intersection. Inversionalso commutes with unary operation of complementation as well as with taking suprema and infima. Inversion is alsocompatible with the ordering of relations by inclusion.[1]

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its inverse is too.

6.3 See also• Bijection

• Function (mathematics)

• Inverse function

• Relation (mathematics)

• Transpose graph

6.4 References[1] Gunther Schmidt; Thomas Ströhlein (1993). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer

Berlin Heidelberg. pp. 9–10. ISBN 978-3-642-77970-1.

[2] Celestina Cotti Ferrero; Giovanni Ferrero (2002). Nearrings: Some Developments Linked to Semigroups and Groups.Kluwer Academic Publishers. p. 3. ISBN 978-1-4613-0267-4.

[3] Joachim Lambek (2001). “Relations Old and New”. In Ewa Orlowska, Andrzej Szalas. Relational Methods for ComputerScience Applications. Springer Science & Business Media. pp. 135–146. ISBN 978-3-7908-1365-4.

• Halmos, Paul R. (1974), Naive Set Theory, p. 40, ISBN 978-0-387-90092-6

Chapter 7

Partially ordered set

{x,y,z}

{y,z}{x,z}{x,y}

{y} {z}{x}

Ø

The Hasse diagram of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Sets on the same horizontal leveldon't share a precedence relationship. Other pairs, such as {x} and {y,z}, do not either.

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitiveconcept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together witha binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other.Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for somepairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiartotal orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depictsthe ordering relation.[1]

A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy.Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.

24

7.1. FORMAL DEFINITION 25

7.1 Formal definition

A (non-strict) partial order[2] is a binary relation "≤" over a set P which is reflexive, antisymmetric, and transitive,i.e., which satisfies for all a, b, and c in P:

• a ≤ a (reflexivity);

• if a ≤ b and b ≤ a then a = b (antisymmetry);

• if a ≤ b and b ≤ c then a ≤ c (transitivity).

In other words, a partial order is an antisymmetric preorder.A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimesalso used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered setscan also be referred to as “ordered sets”, especially in areas where these structures are more common than posets.For a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they areincomparable. In the figure on top-right, e.g. {x} and {x,y,z} are comparable, while {x} and {y} are not. A partialorder under which every pair of elements is comparable is called a total order or linear order; a totally orderedset is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no twodistinct elements are comparable is called an antichain (e.g. the set of singletons {{x}, {y}, {z}} in the top-rightfigure). An element a is said to be covered by another element b, written a<:b, if a is strictly less than b and no thirdelement c fits between them; formally: if both a≤b and a≠b are true, and a≤c≤b is false for each c with a≠c≠b. Amore concise definition will be given below using the strict order corresponding to "≤". For example, {x} is coveredby {x,z} in the top-right figure, but not by {x,y,z}.

7.2 Examples

Standard examples of posets arising in mathematics include:

• The real numbers ordered by the standard less-than-or-equal relation ≤ (a totally ordered set as well).

• The set of subsets of a given set (its power set) ordered by inclusion (see the figure on top-right). Similarly, theset of sequences ordered by subsequence, and the set of strings ordered by substring.

• The set of natural numbers equipped with the relation of divisibility.

• The vertex set of a directed acyclic graph ordered by reachability.

• The set of subspaces of a vector space ordered by inclusion.

• For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequencea precedes sequence b if every item in a precedes the corresponding item in b. Formally, (an)n∈ℕ ≤ (bn) ∈ℕif and only if a ≤ b for all n in ℕ, i.e. a componentwise order.

• For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ gif and only if f(x) ≤ g(x) for all x in X.

• A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d ...

7.3 Extrema

There are several notions of “greatest” and “least” element in a poset P, notably:

26 CHAPTER 7. PARTIALLY ORDERED SET

• Greatest element and least element: An element g in P is a greatest element if for every element a in P, a ≤ g.An element m in P is a least element if for every element a in P, a ≥ m. A poset can only have one greatest orleast element.

• Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a inP such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a <m. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be morethan one maximal element, and similarly for least elements and minimal elements.

• Upper and lower bounds: For a subset A of P, an element x in P is an upper bound of A if a ≤ x, for eachelement a in A. In particular, x need not be in A to be an upper bound of A. Similarly, an element x in P is alower bound of A if a ≥ x, for each element a in A. A greatest element of P is an upper bound of P itself, anda least element is a lower bound of P.

For example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements;on the other hand this poset does not have a greatest element (although if one would include 0 in the poset, whichis a multiple of any integer, that would be a greatest element; see figure). This partially ordered set does not evenhave any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal. If thenumber 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting posetdoes not have a least element, but any prime number is a minimal element for it. In this poset, 60 is an upper bound(though not a least upper bound) of the subset {2,3,5,10}, which does not have any lower bound (since 1 is not in theposet); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound.

7.4 Orders on the Cartesian product of partially ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesianproduct of two partially ordered sets are (see figures):

• the lexicographical order: (a,b) ≤ (c,d) if a < c or (a = c and b ≤ d);

• the product order: (a,b) ≤ (c,d) if a ≤ c and b ≤ d;

• the reflexive closure of the direct product of the corresponding strict orders: (a,b) ≤ (c,d) if (a < c and b < d)or (a = c and b = d).

All three can similarly be defined for the Cartesian product of more than two sets.Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space.See also orders on the Cartesian product of totally ordered sets.

7.5 Sums of partially ordered sets

Another way to combine two posets is the ordinal sum[3] (or linear sum[4]), Z = X ⊕ Y, defined on the union of theunderlying sets X and Y by the order a ≤Z b if and only if:

• a, b ∈ X with a ≤X b, or

• a, b ∈ Y with a ≤Y b, or

• a ∈ X and b ∈ Y.

If two posets are well-ordered, then so is their ordinal sum.[5]

7.6. STRICT AND NON-STRICT PARTIAL ORDERS 27

7.6 Strict and non-strict partial orders

In some contexts, the partial order defined above is called a non-strict (or reflexive, orweak) partial order. In thesecontexts a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive, transitive and asymmetric,i.e. which satisfies for all a, b, and c in P:

• not a < a (irreflexivity),

• if a < b and b < c then a < c (transitivity), and

• if a < b then not b < a (asymmetry; implied by irreflexivity and transitivity[6]).

There is a 1-to-1 correspondence between all non-strict and strict partial orders.If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the irreflexive kernel given by:

a < b if a ≤ b and a ≠ b

Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the reflexive closuregiven by:

a ≤ b if a < b or a = b.

This is the reason for using the notation "≤".Using the strict order "<", the relation "a is covered by b" can be equivalently rephrased as "a<b, but not a<c<b forany c". Strict partial orders are useful because they correspond more directly to directed acyclic graphs (dags): everystrict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself.

7.7 Inverse and order dual

The inverse or converse ≥ of a partial order relation ≤ satisfies x≥y if and only if y≤x. The inverse of a partialorder relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of apartially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > isto ≥ as < is to ≤.Any one of the four relations ≤, <, ≥, and > on a given set uniquely determines the other three.In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to eachother: either x < y, or x = y, or x > y, or x and y are incomparable (none of the other three). A totally ordered set is onethat rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. Thenatural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitudewhereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; wecould for example order them lexicographically via x+iy < u+iv if and only if x < u or (x = u and y < v), but this is notordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitudeyields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolutemagnitude but are not equal, violating antisymmetry.

7.8 Mappings between partially ordered sets

Given two partially ordered sets (S,≤) and (T,≤), a function f: S → T is called order-preserving, or monotone,or isotone, if for all x and y in S, x≤y implies f(x) ≤ f(y). If (U,≤) is also a partially ordered set, and both f: S→ T and g: T → U are order-preserving, their composition (g∘f): S → U is order-preserving, too. A function f:S → T is called order-reflecting if for all x and y in S, f(x) ≤ f(y) implies x≤y. If f is both order-preserving andorder-reflecting, then it is called an order-embedding of (S,≤) into (T,≤). In the latter case, f is necessarily injective,since f(x) = f(y) implies x ≤ y and y ≤ x. If an order-embedding between two posets S and T exists, one says that Scan be embedded into T. If an order-embedding f: S → T is bijective, it is called an order isomorphism, and the

28 CHAPTER 7. PARTIALLY ORDERED SET

partial orders (S,≤) and (T,≤) are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagrams(cf. right picture). It can be shown that if order-preserving maps f: S → T and g: T → S exist such that g∘f and f∘gyields the identity function on S and T, respectively, then S and T are order-isomorphic. [7]

For example, a mapping f: ℕ → ℙ(ℕ) from the set of natural numbers (ordered by divisibility) to the power set ofnatural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. Itis order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. However, it is neitherinjective (since it maps both 12 and 6 to {2,3}) nor order-reflecting (since besides 12 doesn't divide 6). Takinginstead each number to the set of its prime power divisors defines a map g: ℕ → ℙ(ℕ) that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it e.g. doesn't map any number tothe set {4}), but it can be made one by restricting its codomain to g(ℕ). The right picture shows a subset of ℕ and itsisomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to awide class of partial orders, called distributive lattices, see "Birkhoff’s representation theorem".

7.9 Number of partial orders

Partially ordered set of set of all subsets of a six-element set {a, b, c, d, e, f}, ordered by the subset relation.

Sequence A001035 in OEIS gives the number of partial orders on a set of n labeled elements:The number of strict partial orders is the same as that of partial orders.If we count only up to isomorphism, we get 1, 1, 2, 5, 16, 63, 318, … (sequence A000112 in OEIS).

7.10 Linear extension

A partial order ≤* on a set X is an extension of another partial order ≤ on X provided that for all elements x and yof X, whenever x ≤ y, it is also the case that x ≤* y. A linear extension is an extension that is also a linear (i.e., total)order. Every partial order can be extended to a total order (order-extension principle).[8]

In computer science, algorithms for finding linear extensions of partial orders (represented as the reachability ordersof directed acyclic graphs) are called topological sorting.

7.11. IN CATEGORY THEORY 29

7.11 In category theory

Every poset (and every preorder) may be considered as a category in which every hom-set has at most one element.More explicitly, let hom(x, y) = {(x, y)} if x ≤ y (and otherwise the empty set) and (y, z)∘(x, y) = (x, z). Posets areequivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an initialobject, and the largest element, if it exists, is a terminal object. Also, every preordered set is equivalent to a poset.Finally, every subcategory of a poset is isomorphism-closed.

7.12 Partial orders in topological spaces

Main article: Partially ordered space

If P is a partially ordered set that has also been given the structure of a topological space, then it is customary toassume that {(a, b) : a ≤ b} is a closed subset of the topological product space P ×P . Under this assumption partialorder relations are well behaved at limits in the sense that if ai → a , bi → b and ai ≤ bi for all i, then a ≤ b.[9]

7.13 Interval

For a ≤ b, the closed interval [a,b] is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). It contains atleast the elements a and b.Using the corresponding strict relation "<", the open interval (a,b) is the set of elements x satisfying a < x < b (i.e. a< x and x < b). An open interval may be empty even if a < b. For example, the open interval (1,2) on the integers isempty since there are no integers i such that 1 < i < 2.Sometimes the definitions are extended to allow a > b, in which case the interval is empty.The half-open intervals [a,b) and (a,b] are defined similarly.A poset is locally finite if every interval is finite. For example, the integers are locally finite under their natural order-ing. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since e.g. (1,2)≤(1,3)≤(1,4)≤(1,5)≤...≤(2,1).Using the interval notation, the property "a is covered by b" can be rephrased equivalently as [a,b] = {a,b}.This concept of an interval in a partial order should not be confused with the particular class of partial orders knownas the interval orders.

7.14 See also

• antimatroid, a formalization of orderings on a set that allows more general families of orderings than posets

• causal set

• comparability graph

• complete partial order

• directed set

• graded poset

• incidence algebra

• lattice

• locally finite poset

• Möbius function on posets

• ordered group

30 CHAPTER 7. PARTIALLY ORDERED SET

• poset topology, a kind of topological space that can be defined from any poset

• Scott continuity - continuity of a function between two partial orders.

• semilattice

• semiorder

• series-parallel partial order

• stochastic dominance

• strict weak ordering - strict partial order "<" in which the relation “neither a < b nor b < a" is transitive.

• Zorn’s lemma

7.15 Notes[1] Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons.

p. 28. ISBN 0-471-83817-9. Retrieved 27 July 2012. A partially ordered set is conveniently represented by a Hassediagram...

[2] Simovici, Dan A. & Djeraba, Chabane (2008). “Partially Ordered Sets”. Mathematical Tools for Data Mining: Set Theory,Partial Orders, Combinatorics. Springer. ISBN 9781848002012.

[3] Neggers, J.; Kim, Hee Sik (1998), “4.2 Product Order and Lexicographic Order”, Basic Posets, World Scientific, pp. 62–63,ISBN 9789810235895

[4] Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 17-18

[5] P. R. Halmos (1974). Naive Set Theory. Springer. p. 82. ISBN 978-1-4757-1645-0.

[6] 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”.

[7] Davey, B. A.; Priestley, H. A. (2002). “Maps between ordered sets”. Introduction to Lattices and Order (2nd ed.). NewYork: Cambridge University Press. pp. 23–24. ISBN 0-521-78451-4. MR 1902334.

[8] Jech, Thomas (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 0-486-46624-8.

[9] Ward, L. E. Jr (1954). “Partially Ordered Topological Spaces”. Proceedings of the American Mathematical Society 5 (1):144–161. doi:10.1090/S0002-9939-1954-0063016-5

7.16 References• Deshpande, Jayant V. (1968). “On Continuity of a Partial Order”. Proceedings of the American MathematicalSociety 19 (2): 383–386. doi:10.1090/S0002-9939-1968-0236071-7.

• Schröder, Bernd S. W. (2003). Ordered Sets: An Introduction. Birkhäuser, Boston.

• Stanley, Richard P.. Enumerative Combinatorics 1. Cambridge Studies in Advanced Mathematics 49. Cam-bridge University Press. ISBN 0-521-66351-2.

7.17 External links• A001035: Number of posets with n labeled elements in the OEIS

• A000112: Number of posets with n unlabeled elements in the OEIS

Chapter 8

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.

8.1 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.

8.2 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:

31

32 CHAPTER 8. 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”

8.3 Number of reflexive relations

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

8.4. PHILOSOPHICAL LOGIC 33

8.4 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]

8.5 See also

• Binary relation

• Symmetric relation

• Transitive relation

34 CHAPTER 8. REFLEXIVE RELATION

• Coreflexive relation

8.6 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

8.7 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.

8.8 External links• Hazewinkel, Michiel, ed. (2001), “Reflexivity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 9

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.

35

36 CHAPTER 9. SURJECTIVE FUNCTION

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 byNicolas 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 surmeans over or above and relates to the factthat the image of the domain of a surjective function completely covers the function’s codomain.

9.1 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

9.2 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.

9.3 Properties

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

9.3. PROPERTIES 37

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.

9.3.1 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.

38 CHAPTER 9. SURJECTIVE FUNCTION

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).

9.3.2 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

9.4. SEE ALSO 39

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.

9.3.3 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.

9.3.4 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.

9.3.5 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.

9.3.6 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(~).

9.4 See also• Bijection, injection and surjection

• Cover (algebra)

• Covering map

• Enumeration

• Fiber bundle

• Index set

• Section (category theory)

40 CHAPTER 9. SURJECTIVE FUNCTION

9.5 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.

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

Chapter 10

Total order

Inmathematics, 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.

10.1 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.

41

42 CHAPTER 10. TOTAL ORDER

10.2 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. Thedefinition 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.

10.3 Further concepts

10.3.1 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.

10.3.2 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.

10.3. FURTHER CONCEPTS 43

10.3.3 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).

10.3.4 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.

10.3.5 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.

10.3.6 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.

10.3.7 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

44 CHAPTER 10. TOTAL ORDER

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

∪i Ai is defined by

For x, y ∈∪

i∈I Ai , x ≤ y holds if:1. Either there is some i ∈ I with x ≤i y

2. or there are some i < j in I with x ∈ Ai , y ∈ Aj

10.4 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 ofthe 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.

10.5 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]

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

10.7 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

10.8. REFERENCES 45

10.8 References• George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN0-7167-0442-0

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

Chapter 11

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.

11.1 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.

11.2 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.

11.3 See also

• Total order

46

11.4. REFERENCES 47

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

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

Chapter 12

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.

12.1 Formal definition

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

∀a, b, c ∈ X : (aRb ∧ bRc) ⇒ aRc

12.2 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)

12.3 Properties

48

12.4. COUNTING TRANSITIVE RELATIONS 49

12.3.1 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.

12.3.2 Other properties

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

12.3.3 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

12.4 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]

12.5 See also

• Transitive closure

• Transitive reduction

• Intransitivity

• Reflexive relation

• Symmetric relation

• Quasitransitive relation

• Nontransitive dice

• Rational choice theory

50 CHAPTER 12. TRANSITIVE RELATION

12.6 Sources

12.6.1 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"

12.6.2 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.

12.7 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 13

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.

51

52 CHAPTER 13. 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.

13.1 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]

13.2 Axiomatizations

13.2.1 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).

13.2. AXIOMATIZATIONS 53

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).

13.2.2 Total preorders

Strict weak orders are very closely related to total preorders or (non-strict) weak orders, and the samemathematicalconcepts 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.

13.2.3 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.

54 CHAPTER 13. WEAK ORDERING

13.2.4 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.

13.3 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.

13.4 All weak orders on a finite set

13.4.1 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.

13.4. ALL WEAK ORDERS ON A FINITE SET 55

(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.

13.4.2 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.

56 CHAPTER 13. WEAK ORDERING

13.5 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]

13.6 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, CambridgeUniversity Press, p. 204, ISBN9780521675994.

[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.

13.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 57

13.7 Text and image sources, contributors, and licenses

13.7.1 Text• Asymmetric relation Source: https://en.wikipedia.org/wiki/Asymmetric_relation?oldid=653233105Contributors: Patrick, CharlesMatthews,

Dcoetzee, Brianjd, Paul August, Oliphaunt, Palica, Fresheneesz, Mditto, Arthur Rubin, SmackBot, Jdthood, Tsca.bot, Jklin, Lakinekaki,CRGreathouse, Gregbard, Alastair Haines, Magioladitis, Robin S, Jy00912345, Taifunbrowser, VolkovBot, Henry Delforn (old), Ben-der2k14, Addbot, Yobot, Erik9bot, WikitanvirBot, Theophil789, TheodoreYou, MerlIwBot, Helpful Pixie Bot, Lerutit and Anonymous:12

• Bijection Source: https://en.wikipedia.org/wiki/Bijection?oldid=678287673Contributors: DamianYerrick, AxelBoldt, Tarquin, Jan Hid-ders, XJaM, Toby Bartels, Michael Hardy, Wshun, TakuyaMurata, GTBacchus, Karada, Александър, Glenn, Poor Yorick, Rob Hooft,Pizza Puzzle, Hashar, Hawthorn, CharlesMatthews, Dcoetzee, Dysprosia, Hyacinth, David Shay, Ed g2s, Bevo, Robbot, Fredrik, Benwing,Bkell, Salty-horse, Tobias Bergemann, Giftlite, Jorge Stolfi, Alberto da Calvairate~enwiki, MarkSweep, Tsemii, Vivacissamamente, Gua-nabot, Guanabot2, Quistnix, Paul August, Ignignot, MisterSheik, Nickj, Kevin Lamoreau, Obradovic Goran, Pearle, HasharBot~enwiki,Dallashan~enwiki, ABCD, Schapel, Palica, MarSch, Salix alba, FlaBot, VKokielov, RexNL, Chobot, YurikBot, Michael Slone, Member,SmackBot, RDBury, Mmernex, Octahedron80, Mhym, Bwoodacre, Dreadstar, Davipo, Loadmaster, Mets501, Dreftymac, Hilverd, John-fuhrmann, Bill Malloy, Domitori, JRSpriggs, CmdrObot, Gregbard, Yaris678, Sam Staton, Panzer raccoon!, Kilva, AbcXyz, Escarbot,Salgueiro~enwiki, JAnDbot, David Eppstein, Martynas Patasius, Paulnwatts, Cpiral, GaborLajos, Policron, Diegovb, UnicornTapestry,Yomcat, Wykypydya, Bongoman666, SieBot, Paradoctor, Paolo.dL, Smaug123, MiNombreDeGuerra, JackSchmidt, I Spel Good~enwiki,Peiresc~enwiki, Classicalecon, Adrianwn, Biagioli, Watchduck, Hans Adler, Humanengr, Neuralwarp, Baudway, FactChecker1199, Kal-El-Bot, Subversive.sound, Tanhabot, Glane23, PV=nRT, Meisam, Legobot, Luckas-bot, Yobot, Ash4Math, Shvahabi, Omnipaedista,RibotBOT, Thehelpfulbot, FrescoBot, MarcelB612, CodeBlock, MastiBot, FoxBot, Duoduoduo, Xnn, EmausBot, Hikaslap, TuHan-Bot,Cobaltcigs, Wikfr, Karthikndr, Anita5192, Wcherowi, Widr, Strike Eagle, PhnomPencil, Knwlgc, Dhoke sanket, Victor Yus, Dexbot,Cerabot~enwiki, JPaestpreornJeolhlna, Yardimsever, CasaNostra, KoriganStone, Whamsicore, JMP EAX, Sweepy and Anonymous: 89

• Binary relation Source: https://en.wikipedia.org/wiki/Binary_relation?oldid=678287658 Contributors: AxelBoldt, Bryan Derksen, Zun-dark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64, Takuya-Murata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Fropuff,Dratman, Jorge Stolfi, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, PAR, Adrian.benko, OlegAlexandrov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koffieyahoo, Trovatore,Bota47, Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, Jon Awbrey, Turms, Lambiam, Dbtfz, Mr Stephen,Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egriffin, Rlupsa, JAnD-bot, MER-C, Magioladitis, Vanish2, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr, Policron, DavidCBryant,Quux0r, VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq, AlleborgoBot, AHMartin,Ocsenave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot, CBM2, Classicalecon,ClueBot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jarble, Legobot, Luckas-bot,Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote, Hahahaha4, Materialscientist,Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard McCay, Constructive editor, Mark Re-nier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Nomen4Omen,Chharvey, SporkBot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi, Frietjes, Helpful Pixie Bot, Ko-ertefa, ChrisGualtieri, YFdyh-bot, Dexbot, Makecat-bot, Lerutit, Jochen Burghardt, Jodosma, Karim132, Monkbot, Pratincola, ,Some1Redirects4You and Anonymous: 102

• Composition of relations Source: https://en.wikipedia.org/wiki/Composition_of_relations?oldid=644804909 Contributors: Rp, AugPi,Charles Matthews, Giftlite, EmilJ, Oliphaunt, MFH, SixWingedSeraph, MarSch, Nbarth, Lambiam, Happy-melon, CBM, Sam Staton,David Eppstein, Synthebot, Classicalecon, Hans Adler, Addbot, Luckas-bot, Yobot, Pcap, FrescoBot, Gamewizard71, Quondum, AgileAntechinus, JMP EAX and Anonymous: 6

• Demonic composition Source: https://en.wikipedia.org/wiki/Demonic_composition?oldid=633771432 Contributors: Michael Hardy,David Eppstein, LokiClock, Classicalecon, AnomieBOT and Anonymous: 2

• Inverse relation Source: https://en.wikipedia.org/wiki/Inverse_relation?oldid=657149498 Contributors: Patrick, Charles Matthews, Al-tenmann, Tobias Bergemann, Giftlite, Kaldari, Spiffy sperry, El C, Versageek, Jeffrey O. Gustafson, MFH, YurikBot, Paul Erik, Knowled-geOfSelf, C.Fred, Nbarth, Faaaa, Nixeagle, Jon Awbrey, Mearnhardtfan, Coredesat, Slakr, DBooth, CBM, Thomasmeeks, Gogo Dodo,Mojo Hand, Hut 8.5, MetsBot, David Eppstein, Real World Apple, Ars Tottle, VolkovBot, Iamthedeus, Maelgwnbot, Classicalecon,Rjd0060, Overstay, Trainshift, Pluto Car, Unco Guid, Viva La Information Revolution!, Autocratic Uzbek, Poke Salat Annie, FlowerMound Belle, Navy Pierre, Mrs. Lovett’s Meat Puppets, Chester County Dude, Southeast Penna Poppa, Delaware Valley Girl, Myst-Bot, Omphaloskeptor, Addbot, Quercus solaris, Luckas-bot, Yobot, Pcap, Materialscientist, MathHisSci, ویکی ,علی John of Reading,SporkBot, Mark viking, JMP EAX and Anonymous: 12

• Partially ordered set Source: https://en.wikipedia.org/wiki/Partially_ordered_set?oldid=672760099 Contributors: Bryan Derksen, Zun-dark, Tomo, Patrick, Bcrowell, Chinju, TakuyaMurata, GTBacchus, AugPi, Charles Matthews, Timwi, Dcoetzee, Dysprosia, Doradus,Maximus Rex, Fibonacci, Tobias Bergemann, Giftlite, Markus Krötzsch, Fropuff, Peruvianllama, Jason Quinn, Neilc, Gubbubu, De-fLog~enwiki, MarkSweep, Urhixidur, TheJames, Paul August, Zaslav, Spoon!, Porton, Haham hanuka, DougOrleans, Msh210, OlegAlexandrov, Daira Hopwood, MFH, Salix alba, FlaBot, Vonkje, Chobot, Laurentius, Dmharvey, Vecter, JosephSilverman, Sanguinity,Modify, RDBury, IncnisMrsi, Brick Thrower, Cesine, Zanetu, Jcarroll, Nbarth, Jdthood, Javalenok, Kjetil1001, Dreadstar, Esoth~enwiki,Mike Fikes, A. Pichler, Vaughan Pratt, CRGreathouse, L'œuf, CBM, Werratal, Rlupsa, CZeke, Ill logic, JAnDbot, MER-C, BrotherE,Tbleher, A3nm, David Eppstein, SlamDiego, Bissinger, Haseldon, Daniel5Ko, GaborLajos, NewEnglandYankee, Orphic, RobertDanielE-merson, TXiKiBoT, Digby Tantrum, PaulTanenbaum, Arcfrk, SieBot, Mochan Shrestha, TheGhostOfAdrianMineha, Thehotelambush,Megaloxantha, Peiresc~enwiki, Cheesefondue, Jludwig, ClueBot, Morinus, Justin W Smith, Methossant, Pi zero, Jonathanrcoxhead,Watchduck, ComputerGeezer, He7d3r, Hans Adler, Jtle515, Palnot, Marc van Leeuwen, Ankan babee, Addbot, Download, Luckyz,Legobot, Kilom691, AnomieBOT, Erel Segal, Citation bot, SteveWoolf, Undsoweiter, FrescoBot, Nicolas Perrault III, Confluente, Ri-cardo Ferreira de Oliveira, Throw it in the Fire, Gnathan87, Setitup, EmausBot, John of Reading, Febuiles, Thecheesykid, ZéroBot,

58 CHAPTER 13. WEAK ORDERING

Chharvey, The man who was Friday, SporkBot, Zfeinst, Rathgemz, CocuBot, Vdamanafshan, Mesoderm, MerlIwBot, Wbm1058, Jak-shap, Paolo Lipparini, ElphiBot, Larion Garaczi, Aabhis, Jochen Burghardt, Mark viking, Eamonford, Sgbmyr, K401sTL3, Tudor987,Victor Lesyk, Some1Redirects4You and Anonymous: 79

• 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

• 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

• 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 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, ClueBotNG, Pars99, Sourabh.khot, Justincheng12345-bot, Lerutit, Loraof and Anonymous: 61

• 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

13.7.2 Images• File:13-Weak-Orders.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3c/13-Weak-Orders.svg License: Public do-

main Contributors: Transferred from en.wikipedia to Commons. Original artist: SVG version created by Jafet.vixle at en.wikipedia,based on a public-domain PNG version created 2006-0 9-14 by David Eppstein

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