Demonic Composition

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Demonic composition
From Wikipedia, the free encyclopedia

Contents
1

2

3

Asymmetric relation

1

1.1

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.3

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.4

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Bijection

3

2.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2.1

Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2.2

Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.3

More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . .

5

2.4

Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.5

Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.6

Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.7

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.8

Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.9

Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Binary relation

9

3.1

9

Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1

Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.1.2

Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.2.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.3

Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.4

Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.4.1

14

3.2

Difunctional

Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i

ii

4

5

6

7

CONTENTS
3.4.2

Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.4.3

Algebras, categories, and rewriting systems

. . . . . . . . . . . . . . . . . . . . . . . . .

15

3.5

Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.6

The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.7

Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.8

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.9

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

3.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Composition of relations

19

4.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.2

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.3

Join: another form of composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

4.4

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

4.5

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

4.6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Demonic composition

21

5.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

5.2

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Inverse relation

22

6.1

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

6.1.1

Inverse relation of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

6.2

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

6.3

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

6.4

References

23

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Partially ordered set

24

7.1

Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

7.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

7.3

Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

7.4

Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . .

26

7.5

Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

7.6

Strict and non-strict partial orders

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

7.7

Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

7.8

Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

7.9

Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

7.10 Linear extension

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

7.11 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

7.12 Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

7.13 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

CONTENTS

8

9

iii

7.14 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

7.15 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

7.16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

7.17 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

Reflexive relation

31

8.1

Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

8.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

8.3

Number of reflexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

8.4

Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

8.5

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

8.6

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

8.7

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

8.8

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Surjective function

35

9.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

9.2

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

9.3

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

9.3.1

Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

9.3.2

Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

9.3.3

Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

9.3.4

Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

9.3.5

Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

9.3.6

Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

9.4

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

9.5

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

9.6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

10 Total order

41

10.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

10.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

10.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

10.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

10.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

10.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

10.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

10.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

10.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

10.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

10.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . .

44

10.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

iv

CONTENTS
10.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

10.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

10.8 References

45

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 Total relation

46

11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

11.2 Properties and related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

11.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

11.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

12 Transitive relation

48

12.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

12.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

12.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

12.3.1 Closure properties
12.3.2 Other properties

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49

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

12.3.3 Properties that require transitivity

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

12.4 Counting transitive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

12.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

12.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

12.6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

12.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

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

50

13 Weak ordering

51

13.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

13.2 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

13.2.1 Strict weak orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

13.2.2 Total preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

13.2.3 Ordered partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

13.2.4 Representation by functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

13.3 Related types of ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

13.4 All weak orders on a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

13.4.1 Combinatorial enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

13.4.2 Adjacency structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

13.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

13.6 References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

13.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . .

57

13.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

13.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13.7.3 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 characterizing
the 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, every
transitive asymmetric relation is a strict partial order.
Not all asymmetric relations are strict partial orders, however. An example of an asymmetric intransitive relation is
the 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 ≤ x
and both are true. In general, any relation in which x R x holds for some x (that is, which is not irreflexive) is also not
asymmetric.
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 < from
the 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
1

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

Chapter 2

Bijection

X

Y

1

D

2

B

3

C

4

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 elements
of two sets, where every element of one set is paired with exactly one element of the other set, and every element
of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical
3

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 set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence
of a bijection means they have the same number of elements. For infinite sets the picture is more complicated, leading
to 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 to many areas of mathematics 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 see
properties (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 injective
functions).[1] With this terminology, a bijection is a function which is both a surjection and an injection, or using
other 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 the
nine 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 in
the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that
for each position in the order, there is some player batting in that position and property (4) states that two or more
players 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 them
all to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set
of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor
observed 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 count
either 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 a
bijection. 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 paired
with 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 numbers
R+ ≡ (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 oneto-one (injective). However, if the domain is restricted to R+
0 ≡ [0, +∞) , then h becomes bijective; its inverse
is the positive square root function.

2.4 Inverses
A bijection f with domain X (“functionally” indicated by f: X → Y) also defines a relation starting in Y and going to X
(by turning the arrows around). The process of “turning the arrows around” for an arbitrary function does not usually
yield 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 inverse
function exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function is
invertible 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 of
one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has
an inverse function which takes as input a position in the batting order and outputs the player who will be batting in
that 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 have
the same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of “same number of
elements” (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

X

Y

Z

1

D

P

2

B

Q

3

C

R

A
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 the
codomain 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 of
bijections from S to S. That is to say, the number of permutations of elements of S is the same as the number
of 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 not
always the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphisms
must be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphisms
which 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) partial
function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse
to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is
called 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 turns
out 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 A
and likewise B′ ⊆B.[3]
When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[4] An
example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended
complex plane.[5]

2.10 Contrast with
This 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 University
Press. 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 978-0-82479662-4.
[5] John Meakin (2007). “Groups and semigroups: connections and contrasts”. In C.M. Campbell, M.R. Quick, E.F. Robertson, 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). “The Möbius Inverse Monoid”. 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 may
be 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 978-1-55608010-4
• Weisstein, Eric W., “Bijection”, MathWorld.
• Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history
of Injection and related terms.

Chapter 3

Binary relation
“Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For a more
combinatorial 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 a
subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset
of 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 every
prime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). In
this relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and the
prime 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 linear
algebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations are
also 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 the
jth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation on
Z×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. This
extension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in set
theory, 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), and
G is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) and
codomain (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 corresponds
to viewing R as the characteristic function on X × Y for the set of pairs of G.
The order of the elements in each pair of G is important: if a ≠ b, then aRb and bRa can be true or false, independently
of 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 the
relation from X to Y is the subset G of X × Y, and “from X to Y" must always be either specified or implied by the
context 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 codomains
are considered different. For example, if G = {(1, 2), (1, 3), (2, 7)} , then (Z, Z, G) , (R, N, G) , and (N, R, G) are
three 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 with
their graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least one
y 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 distinguishing between a function’s codomain and its range. Thus, a single “rule,” like mapping every real number x to
x2 , can lead to distinct functions f : R → R and f : R → R+ , depending on whether the images under that
rule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets of
ordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As an
example, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees it
as 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 two
definitions 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}. Suppose
that 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 ordered
pairs 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 Y
can 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. For
example, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5
and 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 Y
it holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations in
the picture are functional. An example for a non-functional relation can be obtained by rotating the red graph
clockwise 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 function
or a multivalued function. Note that this property, although sometimes also referred to as total, is different
from 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 green
relation 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 as
one-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 that
R=RR−1 R.[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 predecessor
neighborhood 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 x1 R and x2 R have a
non-empty intersection, then these two sets coincide; formally x1 R ∩ x2 R ≠ ∅ implies x1 R = x2 R.[11]
As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If one
considers a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalence
relation), 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 → C
and 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 functions
f: 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 terminology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctional
relation 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 computer
science, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations are
widely 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 the
involution 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 the
relation on integers in which each odd number is related to itself and there are no other relations. The equality
relation 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 the
above 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 >, but
only 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 both
anti-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 and
only 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 total
in 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 trichotomous
relation, 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 if
x=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. But
it is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the natural
numbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rational
numbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x. A serial relation can
be equivalently characterized as every element having a non-empty successor neighborhood (see the previous
section for the definition of this notion). Similarly an inverse serial relation is a relation in which every element
has 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 relations
on 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 being
reflexive) 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 called
a total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least element
is 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 relation
over 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 exists
y ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. The order of R and S in the notation S ∘ R, used here agrees with
the standard notational order for composition of functions. For example, the composition “is mother of” ∘ “is
parent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “is
grandmother 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 y
always implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is contained
in ≥.
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 its
inverse if and only if it is symmetric. See also duality (order theory). For example, “is less than” (<) is the
inverse 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 containing
R. This can be proven to be equal to the intersection of all reflexive relations containing R.
• Reflexive reduction: R ≠ , defined as R
contained in R.



= R \ { (x, x) | x ∈ X } or the largest irreflexive relation over X

• Transitive closure: R + , defined as the smallest transitive relation over X containing R. This can be seen to be
equal 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 containing
R.

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 and
y are in S.
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial
order, 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 general
not equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother of
the woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, the
transitive closure of “is parent of” is “is ancestor of"; its restriction to females does relate a woman with her paternal
grandmother.

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 an
upper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers this
supremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to the
set 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 relation
algebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finite
and 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 abstract
rewriting system.

3.5 Sets versus classes
Certain mathematical “relations”, such as “equal to”, “member of”, and “subset of”, cannot be understood to be binary
relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of
axiomatic set theory. For example, if we try to model the general concept of “equality” as a binary relation =, we
must 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 because
they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem
is 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 of
a specific set A): the resulting set relation can be denoted ⊆A. Also, the “member of” relation needs to be restricted
to have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown that
assuming ∈ 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 of
the ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course one
can identify the function with its graph in this context.)[24] With this definition one can for instance define a function
relation between every set and its power set.

3.6 The number of binary relations
2

The number of distinct binary relations on an n-element set is 2n (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 neither
a 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 own
complement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse complement).

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. ISBN
0-486-61630-4.
[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in
1996 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; Panos M. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science & Business
Media. 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. ISBN
978-1-4020-6164-6.
[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 978-0-13460643-9.
• Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the High
Level 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 International
Conference, COSIT 2007, Melbourne, Australiia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science
4736, 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 Computer
Science 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 978-0-82186588-0.
[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. Springer
Science & Business Media. p. 496. ISBN 978-3-540-67995-0.
[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), A Transition to Advanced Mathematics (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: School
of Mathematics – Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as “strictly
antisymmetric”.
[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 and
Graphs, 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 978-155608-010-4

Chapter 4

Composition of relations
In mathematics, the composition of binary relations is a concept of forming a new relation S ∘ R from two given
relations 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 element
y ∈ Y such that x R 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 is
such that function composition (with the usual notation) is obtained as a special case, when R and S are functional
relations. Some authors[1] prefer to write ◦l and ◦r explicitly when necessary, depending whether the left or the right
relation 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 use
of semicolon coincides with the notation for function composition used (mostly by computer scientists) in Category
theory,[4] as well as the notation for dynamic conjunction within linguistic dynamic semantics.[5] The semicolon
notation (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 which
has the sets as objects. In Rel, composition of morphisms is exactly composition of relations as defined above. The
category Set of sets is a subcategory of Rel that has the same objects but fewer morphisms. A generalization of this
is 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 a
semigroup 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 composition
forms 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, are
found in the join operation of relational algebra. The usual composition of two binary relations as defined here can
be obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middle
component.

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 and
Graphs, 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 ordinary
composition 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
Suppose R is a binary relation between X and Y and S is a relation between Y and Z. Their right demonic composition
R ;→ S is a relation between X and Z. Its graph is defined as

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

5.2 References
• Backhouse, Roland; van der Woude, Jaap (1993), “Demonic operators and monotype factors”, Mathematical
Structures 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 is
switched in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms,
if X 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 and
only if x L y . 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 inverse
in 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 a
dagger category on the category of relations as detailed below. As a unary operation, taking the inverse (sometimes
called inversion) commutes however with the order-related operations of relation algebra, i.e. it commutes with
union, intersection, complement etc.
The inverse relation is also called the converse relation or transpose relation— the latter in view of its similarity
with the transpose of a matrix.[1] It has also been called the opposite or dual of the original relation.[2] Other notations
˘ or L° or L∨ .
for the inverse relation include LC , LT , 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 inverse
function.
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. This
condition is sufficient for f −1 being a partial function, and it is clear that f −1 then is a (total) function if and only if
f 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 of
relations), 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 relations Rel), 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 dagger
categories, it is self-adjoint.
Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relations
as sets), and actually an involutive quantale. Similarly, the category of heterogenous relations, Rel is also an ordered
category.[3]
In relation algebra (which is an abstraction of the properties of the algebra of endorelations on a set), inversion (the
operation of taking the inverse relation) commutes with other binary operations of union and intersection. Inversion
also commutes with unary operation of complementation as well as with taking suprema and infima. Inversion is also
compatible with the ordering of relations by inclusion.[1]
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial
order, 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 Computer
Science 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}

{x,y}

{x,z}

{y,z}

{x}

{y}

{z}

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 level
don'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 intuitive
concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with
a 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 some
pairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiar
total orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depicts
the 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 sometimes
also used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered sets
can 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 are
incomparable. In the figure on top-right, e.g. {x} and {x,y,z} are comparable, while {x} and {y} are not. A partial
order under which every pair of elements is comparable is called a total order or linear order; a totally ordered
set is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no two
distinct elements are comparable is called an antichain (e.g. the set of singletons {{x}, {y}, {z}} in the top-right
figure). An element a is said to be covered by another element b, written a<:b, if a is strictly less than b and no third
element 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. A
more concise definition will be given below using the strict order corresponding to "≤". For example, {x} is covered
by {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, the
set 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 sequence
a precedes sequence b if every item in a precedes the corresponding item in b. Formally, (an)n∈ℕ ≤ (bn) ∈ℕ
if 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 ≤ g
if 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 or
least element.
• Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a in
P 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 more
than 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 each
element 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 a
lower bound of A if a ≥ x, for each element a in A. A greatest element of P is an upper bound of P itself, and
a 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, which
is a multiple of any integer, that would be a greatest element; see figure). This partially ordered set does not even
have any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal. If the
number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset
does 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 the
poset); 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 Cartesian
product 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 the
underlying 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, or weak) partial order. In these
contexts 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 closure
given 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 for
any c". Strict partial orders are useful because they correspond more directly to directed acyclic graphs (dags): every
strict 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 partial
order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of a
partially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > is
to ≥ 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 each
other: 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 one
that rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. The
natural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitude
whereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; we
could 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 not
ordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitude
yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolute
magnitude 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 and
order-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 S
can 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∘g
yields 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 of
natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. It
is order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. However, it is neither
injective (since it maps both 12 and 6 to {2,3}) nor order-reflecting (since besides 12 doesn't divide 6). Taking
instead each number to the set of its prime power divisors defines a map g: ℕ → ℙ(ℕ) that is order-preserving, orderreflecting, and hence an order-embedding. It is not an order-isomorphism (since it e.g. doesn't map any number to
the set {4}), but it can be made one by restricting its codomain to g(ℕ). The right picture shows a subset of ℕ and its
isomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to a
wide 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 y
of 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 orders
of 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 are
equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an initial
object, 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 to
assume that {(a, b) : a ≤ b} is a closed subset of the topological product space P × P . Under this assumption partial
order 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 at
least 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 is
empty 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 ordering. 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 known
as 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 Hasse
diagram...
[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: School
of Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relations
as “strictly antisymmetric”.
[7] Davey, B. A.; Priestley, H. A. (2002). “Maps between ordered sets”. Introduction to Lattices and Order (2nd ed.). New
York: 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 Mathematical
Society 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. Cambridge 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 other
words, 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 is
equal 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. An
example is the “greater than” relation (x>y) on the real numbers. Note that not every relation which is not reflexive
is 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 of
even 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 of
real numbers: not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the same
limit 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 reflexive
closure 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 equivalent
to 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 2n

2

−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 the
mathematical 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 — A Modern Introduction. Wadsworth. ISBN
1-133-05000-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. ISBN
0-674-55451-5
• 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 978-1-55608010-4

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

X

Y

1

D

2

B

3

C

4
A surjective function from domain X to codomain Y. The function is surjective because every point in the codomain is the value of
f(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 the
same element of Y.
The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki,[1] the pseudonym
for a group of mainly French 20th-century mathematicians who wrote a series of books presenting an exposition of
modern advanced mathematics, beginning in 1935. The French prefix sur means over or above and relates to the fact
that 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 domain
X and codomain Y is surjective if for every y in Y there exists at least one x in X with f (x) = y . Surjections are
sometimes 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 y
we 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 the
solution set of the cubic polynomial equation x3 − 3x − y = 0 and every cubic polynomial with real coefficients has at
least one real root. However, this function is not injective (and hence not bijective) since e.g. the pre-image of y = 2
is {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 y
in 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 positive
real numbers to the set of all real numbers. Its inverse, the exponential function, is not surjective as its range is the set
of positive real numbers and its domain is usually defined to be the set of all real numbers. The matrix exponential
is not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as a
map from the space of all n×n matrices to the general linear group of degree n, i.e. the group of all n×n invertible
matrices. Under this definition the matrix exponential is surjective for complex matrices, although still not surjective
for 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

f(x)

x

Y

X
f:X→Y

A non-surjective function from domain X to codomain Y. The smaller oval inside Y is the image (also called range) of f. This
function 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, but
rather a relationship between the function and its codomain. Unlike injectivity, surjectivity cannot be read off of the
graph 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 be
undone 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 identity
function on the domain Y of g. The function g need not be a complete inverse of f because the composition in the
other 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 a
right 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 preimage
f −1 (B).
For example, in the first illustration, there is some function g such that g(C) = 4. There is also some function f such
that 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

y

y
y

Y
im f

y

Y

im f

x
x

x

X
x

f :X
Y
y f x

x

X1

f : X1
y

X2

f : X2

Y1

Y2

f x

Interpretation for surjective functions in the Cartesian plane, defined by the mapping f : X → Y, where y = f(x), X = domain of
function, Y = range of function. Every element in the range is mapped onto from an element in the domain, by the rule f. There
may 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 in
X, more than one x can map to the same y. Left: Only one domain is shown which makes f surjective. Right: two possible domains
X1 and X2 are shown.

y

y3
y

y

Y
im f

y2

y

Y
Y
Y

im f
y0

Y

y1

Y

x
x0

X

x

x

X

f :X
Y
y f x

x1

x3

X
x2
x

X1

f : X1
y

X

X
x

Y1

x

X

X2

f : X2

Y2

f x

Non-surjective functions in the Cartesian plane. Although some parts of the function are surjective, where elements y in Y do have
a 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 g
o f = h o f, then g = h. This property is formulated in terms of functions and their composition and can be generalized
to the more general notion of the morphisms of a category and their composition. Right-cancellative morphisms are
called 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 a
morphism 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 X
and Y by identifying it with its function graph. A surjective function with domain X and codomain Y is then a binary
relation 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. (The
proof 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. g
is 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 if
f 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 g
is 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 any
epimorphisms in any category.
Any function can be decomposed into a surjection and an injection: For any function h : X → Z there exist a surjection
f : 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 in
Z. These sets are disjoint and partition X. Then f carries each x to the element of Y which contains it, and g carries
each element of Y to the point in Z to which h sends its points. Then f is surjective since it is a projection map, and
g 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 bijection
defined on a quotient of its domain by collapsing all arguments mapping to a given fixed image. More precisely, every
surjection f : A → B can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalence
classes of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set of
all 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 978-0486-45026-1. 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
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denoted
by infix ≤) on some set X which is transitive, antisymmetric, and total. A set paired with a total order is called a totally
ordered 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 means
that 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. (It
requires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extension
of 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 total
order, 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 whether
we 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 X
by 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, is
itself a total order. For example, any set of words ordered alphabetically is a totally ordered set, viewed as a
subset of a Cartesian product of a countable number of copies of a set formed by adding the space symbol to
the 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, hence
also 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 is
the smallest with a certain property if whenever B has the property, there is an order isomorphism from A to a
subset of B):
• The natural numbers comprise the smallest totally ordered set with no upper bound.
• The integers comprise the smallest totally ordered set with neither an upper nor a lower bound.
• The rational numbers comprise the smallest totally ordered set which is dense in the real numbers. The
definition of density used here says that for every 'a' and 'b' in the real numbers such that 'a' < 'b', there is
a '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 order
topology (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 of
some 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 a
natural number}, where In is the set of natural numbers below n, is a chain in this ordering, as it is totally ordered
under 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

10.3.3

43

Finite total orders

A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subset
thereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observing
that every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphic
to an initial segment of the natural numbers ordered by <. In other words a total order on a set with k elements induces
a bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with order
type 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 maps
which 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, the
order 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 on
N induced by < and the order topology on N induced by > (in this case they happen to be identical but will not in
general).
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 upper
bound. 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 two
points 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 intervals
of 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 is
called 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 of
two totally ordered sets are:
• Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order.
• (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order.
• (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of
the corresponding strict total orders). This is also a partial order.
All three can similarly be defined for the Cartesian product of more than two sets.
Applied to the vector space Rn , each of these make it an ordered vector space.
See also examples of partially ordered sets.
A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding total
preorder 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 orientation
results in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both data
results 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 in
Computing 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), Discrete Mathematics, 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. ISBN
0-7167-0442-0
• 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 is
related 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 either
the 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 less
than” is not a total relation, since one can pick two equal numbers, and then neither the first is less than the second, nor
is 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 of
the 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), A Concise Introduction to Mathematical Logic (3rd ed.), New York: 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 order
relations 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 > C
whenever A ≥ B and B ≥ C, then also A ≥ C
whenever 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 Brenda
is the mother of Claire, then Alice is not the mother of Claire. What is more, it is antitransitive: Alice can never be
the 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

12.3.1

49

Closure properties

The converse of a transitive relation is always transitive: e.g. knowing that “is a subset of” is transitive and “is a
superset 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 first
name 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 name
as” is not generally a transitive relation.
The complement of a transitive relation is not always transitive. For instance, while “equal to” is transitive, “not equal
to” 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) is
known.[2] However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – (sequence A000110 in OEIS), those that are symmetric
and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer[3] has made some progress in this direction, expressing relations with combinations of these
properties 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 relations
as “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 978-1-55608010-4
• 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<c<b

a<b
c<b

c<a<b

c<a
c<b

c<b<a

a<b
a<c

a<c
b<c

a,b,c

b<a
c<a

a<b<c

b<a
b<c

b<a<c

b<c<a
The 13 possible strict weak orderings on a set of three elements {a, b, c}. The only partially ordered sets are coloured, while totally
ordered 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 totally
ordered 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 ordered
sets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at least
one of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of the
elements into disjoint subsets, together with a total order on the subsets). In many cases another representation called
a 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 partition
refinement 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 Maryland
Hunt Cup steeplechase in 2007, The Bruce was the clear winner, but two horses Bug River and Lear Charm tied for
second place, with the remaining horses farther back; three horses did not finish.[3] In the weak ordering describing
this outcome, The Bruce would be first, Bug River and Lear Charm would be ranked after The Bruce but before all
the other horses that finished, and the three horses that did not finish would be placed last in the order but tied with
each other.
The points of the Euclidean plane may be ordered by their distance from the origin, giving another example of a weak
ordering with infinitely many elements, infinitely many subsets of tied elements (the sets of points that belong to a
common circle centered at the origin), and infinitely many points within these subsets. Although this ordering has a
smallest 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 is
better 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 are
within the margin of error of each other. However, if candidate x is statistically tied with y, and y is statistically tied
with 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 is
irreflexive, 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 there
exists 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,c
are incomparable but b and c are related, so incomparability does not form an equivalence relation and this example
is 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 is
incomparable 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 same mathematical
concepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A total
preorder or weak order is a preorder that is total; that is, no pair of items is incomparable. A total preorder ≲ satisfies
the 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 preorders
are 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 strict
weak orders and total preorders in a way that preserves rather than reverses the order of the elements. Thus we take
the inverse of the complement: for a strict weak ordering <, define a total preorder ≲ by setting x ≲ y whenever it is
not the case that y < x. In the other direction, to define a strict weak ordering < from a total preorder ≲ , set x < y
whenever 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 if
x ≲ y and y ≲ x. In the case of a total preorder the corresponding partial order on the set of equivalence classes is a
total order. Two elements are equivalent in a total preorder if and only if they are incomparable in the corresponding
strict 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 a
total order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition[7] and by
Theodore Motzkin a list of sets.[8] An ordered partition of a finite set may be written as a finite sequence of the sets
in 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 a
total ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partition
gives rise to a strict weak ordering in which two elements are incomparable when they belong to the same set in the
partition, and otherwise inherit the order of the sets that contain them.

54

13.2.4

CHAPTER 13. WEAK ORDERING

Representation by functions

For sets of sufficiently small cardinality, a third axiomatization is possible, based on real-valued functions. If X is any
set 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 by
setting 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 realvalued function defined on at least the range of f. Thus e.g. a utility function defines a preference relation. In this
context, 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, there
exist strict weak orders that have no corresponding real function. For example, there is no such function for the
lexicographic order on Rn . Thus, while in most preference relation models the relation defines a utility function up to
order-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 f
induces a strict weak ordering on X by setting a < b if and only if f(a) < f(b). As before, the associated total preorder
is given by setting a ≲ b if and only if f(a) ≲ f(b), and the associated equivalence by setting a ∼ b if and only
if f(a) ∼ f(b). It is not assumed here that f is an injective function, so a class of two equivalent elements on Y
may induce a larger class of equivalent elements on X. Also, f is not assumed to be an surjective function, so a class
of equivalent elements on Y may induce a smaller or empty class on X. However, the function f induces an injective
function that maps the partition on X to that on Y. Thus, in the case of finite partitions, the number of classes in X is
less 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 order
that is trichotomous is called a strict total order.[12] The total preorder which is the inverse of its complement is in
this 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 the
total preorder corresponding to a strict weak order we get a ≲ b and b ≲ a, while in the partial order given by the
reflexive closure we get neither a ≤ b nor b ≤ a. For strict total orders these two associated reflexive relations are
the same: the corresponding (non-strict) total order.[12] The reflexive closure of a strict weak ordering is a type of
series-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-element
set 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 three
ways of partitioning the items into one singleton set and one group of two tied items, and each of these partitions
gives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering is
reversed), 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)

(3,1,2,4)

(4,2,1,3)

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

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

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

(4,2,3,1)

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

(1,3,2,4)

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

(1,2,4,3)

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

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

(1,4,3,2)
The permutohedron on four elements, a three-dimensional convex polyhedron. It has 24 vertices, 36 edges, and 14 two-dimensional
faces, 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 that
add or remove a single order relation to a given ordering. For instance, for three elements, the ordering in which all
three elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strict
weak ordering or total preorder axiomatizations. However, a different kind of move is possible, in which the weak
orderings 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 the
ordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as a
Dedekind 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 moves
that add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the undirected graph that
has 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 the
dichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderings
on the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedron
itself, but not the empty set, as a face). The codimension of a face gives the number of equivalence classes in the
corresponding weak ordering.[14] In this geometric representation the partial cube of moves on weak orderings is the
graph 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 of
combinatorial optimization problem, the prioritization of solutions or of bases is often given by a weak order, determined by a real-valued objective function; the phenomenon of ties in these orderings is called “degeneracy”, and
several types of tie-breaking rule have been used to refine this weak ordering into a total ordering in order to prevent
problems caused by degeneracy.[15]
Weak orders have also been used in computer science, in partition refinement based algorithms for lexicographic
breadth-first search and lexicographic topological ordering. In these algorithms, a weak ordering on the vertices of a
graph (represented as a family of sets that partition the vertices, together with a doubly linked list providing a total
order on the sets) is gradually refined over the course of the algorithm, eventually producing a total ordering that is
the output of the algorithm.[16]
In the Standard Library for the C++ programming language, the set and multiset data types sort their input by a
comparison function that is specified at the time of template instantiation, and that is assumed to implement a strict
weak 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 for
second”, The Baltimore Sun, (subscription required (help)).
[4] Regenwetter, Michel (2006), Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications, Cambridge University Press, pp. 113ff, ISBN 9780521536660.
[5] 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 relations
as “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, Cambridge 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, Encyclopedia of Mathematics and its Applications 7, Addison-Wesley, Theorem 3.1, ISBN 978-0-201-13506-0.
[11] Luce, R. Duncan (1956), “Semiorders and a theory of utility discrimination”, Econometrica 24: 178–191, JSTOR 1905751,
MR 0078632.
[12] Velleman, Daniel J. (2006), How to Prove It: A Structured Approach, Cambridge University Press, p. 204, ISBN 9780521675994.
[13] Eppstein, David; Falmagne, Jean-Claude; Ovchinnikov, Sergei (2008), Media Theory: Interdisciplinary Applied Mathematics, 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 algorithmic
tool 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, ISBN
9780132977739.

13.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

57

13.7 Text and image sources, contributors, and licenses
13.7.1

Text

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CRGreathouse, Gregbard, Alastair Haines, Magioladitis, Robin S, Jy00912345, Taifunbrowser, VolkovBot, Henry Delforn (old), Bender2k14, Addbot, Yobot, Erik9bot, WikitanvirBot, Theophil789, TheodoreYou, MerlIwBot, Helpful Pixie Bot, Lerutit and Anonymous:
12
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Pizza Puzzle, Hashar, Hawthorn, Charles Matthews, Dcoetzee, Dysprosia, Hyacinth, David Shay, Ed g2s, Bevo, Robbot, Fredrik, Benwing,
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Dallashan~enwiki, ABCD, Schapel, Palica, MarSch, Salix alba, FlaBot, VKokielov, RexNL, Chobot, YurikBot, Michael Slone, Member,
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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, KalEl-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,
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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, Oleg
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Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egriffin, Rlupsa, JAnDbot, 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 Renier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Nomen4Omen,
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,
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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, Agile
Antechinus, JMP EAX and Anonymous: 6
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David Eppstein, LokiClock, Classicalecon, AnomieBOT and Anonymous: 2
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Mojo Hand, Hut 8.5, MetsBot, David Eppstein, Real World Apple, Ars Tottle, VolkovBot, Iamthedeus, Maelgwnbot, Classicalecon,
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Mound Belle, Navy Pierre, Mrs. Lovett’s Meat Puppets, Chester County Dude, Southeast Penna Poppa, Delaware Valley Girl, MystBot, Omphaloskeptor, Addbot, Quercus solaris, Luckas-bot, Yobot, Pcap, Materialscientist, MathHisSci, ‫علی ویکی‬, John of Reading,
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58

CHAPTER 13. WEAK ORDERING

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49
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Hardy, Rp, Looxix~enwiki, Andres, Charles Matthews, Dcoetzee, Jitse Niesen, Fredrik, MathMartin, Tobias Bergemann, Giftlite, BenFrantzDale, Gubbubu, Chowbok, Paul August, MyNameIsNotBob, Spoon!, Polluks, DanShearer, Woohookitty, Linas, LOL, Isnow,
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MaratL, Wasseralm, JJL, SmackBot, InverseHypercube, Nbarth, Wen D House, Cybercobra, Jóna Þórunn, Lambiam, Coredesat, Lyonsam, 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, Ottawahitech, Alexbot, Wikibojopayne, Pa68, SilvonenBot, Addbot, Luckas-bot, Yobot, Ptbotgourou, Materialscientist, GrouchoBot, Undsoweiter, RedBot, Katovatzschyn, EmausBot, Slightsmile, IGeMiNix, ChuispastonBot, ClueBot NG, Pars99, Sourabh.khot, Justincheng12345bot, 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 domain 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
• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public domain Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs)
• File:Bijection.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a5/Bijection.svg License: Public domain Contributors:
enwiki Original artist: en:User:Schapel
• File:Bijective_composition.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a2/Bijective_composition.svg License: Public domain Contributors: ? Original artist: ?

13.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

59

• File:Birkhoff120.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7c/Birkhoff120.svg License: Public domain Contributors: Own work Original artist: David Eppstein
• File:Codomain2.SVG Source: https://upload.wikimedia.org/wikipedia/commons/6/64/Codomain2.SVG License: Public domain Contributors: Transferred from en.wikipedia
Original artist: Damien Karras (talk). Original uploader was Damien Karras at en.wikipedia
• File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Original artist: ?
• File:Graph_of_non-injective,_non-surjective_function_(red)_and_of_bijective_function_(green).gif Source: https://upload.wikimedia.
org/wikipedia/commons/b/b0/Graph_of_non-injective%2C_non-surjective_function_%28red%29_and_of_bijective_function_%28green%
29.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: Jochen Burghardt
• File:GreaterThan.png Source: https://upload.wikimedia.org/wikipedia/en/5/59/GreaterThan.png License: Public domain Contributors:
? Original artist: ?
• File:GreaterThanOrEqualTo.png Source: https://upload.wikimedia.org/wikipedia/en/0/0b/GreaterThanOrEqualTo.png License: Public domain Contributors: ? Original artist: ?
• File:Hasse_diagram_of_powerset_of_3.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/ea/Hasse_diagram_of_powerset_
of_3.svg License: CC-BY-SA-3.0 Contributors: self-made using graphviz's dot. Original artist: KSmrq
• File:Hasse_diagram_of_powerset_of_3_no_greatest_or_least.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9e/
Hasse_diagram_of_powerset_of_3_no_greatest_or_least.svg License: CC-BY-SA-3.0 Contributors: Based on File:Hasse diagram of
powerset of 3.svg Original artist: User:Fibonacci, User:KSmrq
• File:Infinite_lattice_of_divisors.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e6/Infinite_lattice_of_divisors.svg
License: Public domain Contributors: w:de:Datei:Verband TeilerN.png Original artist: Watchduck (a.k.a. Tilman Piesk)
• File:Lexicographic_order_on_pairs_of_natural_numbers.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8a/Lexicographic_
order_on_pairs_of_natural_numbers.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Jochen Burghardt
• File:Monotonic_but_nonhomomorphic_map_between_lattices.gif Source: https://upload.wikimedia.org/wikipedia/commons/8/8c/
Monotonic_but_nonhomomorphic_map_between_lattices.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: Jochen
Burghardt
• File:N-Quadrat,_gedreht.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/13/N-Quadrat%2C_gedreht.svg License: CC0
Contributors: Own work Original artist: Mini-floh
• File:Non-surjective_function2.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f0/Non-surjective_function2.svg License: CC BY-SA 3.0 Contributors: http://en.wikipedia.org/wiki/File:Non-surjective_function.svg Original artist: original version: Maschen,
the correction: raffamaiden
• File:Nuvola_apps_edu_mathematics_blue-p.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_
mathematics_blue-p.svg License: GPL Contributors: Derivative work from Image:Nuvola apps edu mathematics.png and Image:Nuvola
apps edu mathematics-p.svg Original artist: David Vignoni (original icon); Flamurai (SVG convertion); bayo (color)
• File:Permutohedron.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Permutohedron.svg License: Public domain
Contributors: Own work Original artist: David Eppstein
• File:Poset6.jpg Source: https://upload.wikimedia.org/wikipedia/en/1/12/Poset6.jpg License: PD Contributors: ? Original artist: ?
• File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0
Contributors:
Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:
Tkgd2007
• File:Strict_product_order_on_pairs_of_natural_numbers.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/88/Strict_
product_order_on_pairs_of_natural_numbers.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Jochen Burghardt
• File:Surjection.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6c/Surjection.svg License: Public domain Contributors: ? Original artist: ?
• File:Surjective_function.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4d/Surjective_function.svg License: Public
domain Contributors: Own work Original artist: Maschen
• File:Venn_A_intersect_B.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6d/Venn_A_intersect_B.svg License: Public domain Contributors: Own work Original artist: Cepheus
• File:Wiki_letter_w_cropped.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1c/Wiki_letter_w_cropped.svg License:
CC-BY-SA-3.0 Contributors:
• Wiki_letter_w.svg Original artist: Wiki_letter_w.svg: Jarkko Piiroinen
• File:Wiktionary-logo-en.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg License: Public
domain Contributors: Vector version of Image:Wiktionary-logo-en.png. Original artist: Vectorized by Fvasconcellos (talk · contribs),
based on original logo tossed together by Brion Vibber

13.7.3

Content license

• Creative Commons Attribution-Share Alike 3.0

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