Binary relation explained

X

and

Y

is a set of ordered pairs

(x,y)

consisting of elements

x

from

X

and

y

from

Y

.[2] It encodes the common concept of relation: an element

x

is related to an element

y

, if and only if the pair

(x,y)

belongs to the set of ordered pairs that defines the binary relation.

An example of a binary relation is the "divides" relation over the set of prime numbers

P

and the set of integers

Z

, in which each prime

p

is related to each integer

z

that is a multiple of

p

, but not to an integer that is not a multiple of

p

. In this relation, for instance, the prime number

2

is related to numbers such as

-4

,

0

,

6

,

10

, but not to

1

or

9

, just as the prime number

3

is related to

0

,

6

, and

9

, but not to

4

or

13

.

Binary relations are used in many branches of mathematics to model a wide variety of concepts. These include, among others:

A function may be defined as a binary relation that meets additional constraints.[3] Binary relations are also heavily used in computer science.

A binary relation over sets

X

and

Y

is an element of the power set of

X x Y.

Since the latter set is ordered by inclusion (

\subseteq

), each relation has a place in the lattice of subsets of

X x Y.

A binary relation is called a homogeneous relation when

X=Y

. A binary relation is also called a heterogeneous relation when it is not necessary that

X=Y

.

Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder,[4] Clarence Lewis,[5] and Gunther Schmidt.[6] A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice.

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.

A binary relation is the most studied special case

n=2

of an

n

-ary relation over sets

X1,...,Xn

, which is a subset of the Cartesian product

X1 x … x Xn.

[2]

Definition

Given sets

X

and

Y

, the Cartesian product

X x Y

is defined as

\{(x,y)\midx\inXandy\inY\},

and its elements are called ordered pairs.

A

R

over sets

X

and

Y

is a subset of

X x Y.

The set

X

is called the or of

R

, and the set

Y

the or of

R

. In order to specify the choices of the sets

X

and

Y

, some authors define a or as an ordered triple

(X,Y,G)

, where

G

is a subset of

X x Y

called the of the binary relation. The statement

(x,y)\inR

reads "

x

is

R

-related to

y

" and is denoted by

xRy

.[4] [5] [6] The or of

R

is the set of all

x

such that

xRy

for at least one

y

. The codomain of definition,, or of

R

is the set of all

y

such that

xRy

for at least one

x

. The of

R

is the union of its domain of definition and its codomain of definition.[7] [8] [9]

When

X=Y,

a binary relation is called a (or). To emphasize the fact that

X

and

Y

are allowed to be different, a binary relation is also called a heterogeneous relation.[10] [11] [12] The prefix hetero is from the Greek ἕτερος (heteros, "other, another, different").

A heterogeneous relation has been called a rectangular relation,[12] suggesting that it does not have the square-like symmetry of a homogeneous relation on a set where

A=B.

Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "... a variant of the theory has evolved that treats relations from the very beginning as or, i.e. as relations where the normal case is that they are relations between different sets."

The terms correspondence,[13] dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product

X x Y

without reference to

X

and

Y

, and reserve the term "correspondence" for a binary relation with reference to

X

and

Y

.

In a binary relation, the order of the elements is important; if

x ≠ y

then

yRx

can be true or false independently of

xRy

. For example,

3

divides

9

, but

9

does not divide

3

.

Operations

Union

If

R

and

S

are binary relations over sets

X

and

Y

then

R\cupS=\{(x,y)\midxRyorxSy\}

is the of

R

and

S

over

X

and

Y

.

The identity element is the empty relation. For example,

\leq

is the union of < and =, and

\geq

is the union of > and =.

Intersection

If

R

and

S

are binary relations over sets

X

and

Y

then

R\capS=\{(x,y)\midxRyandxSy\}

is the of

R

and

S

over

X

and

Y

.

The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2".

Composition

See main article: Composition of relations. If

R

is a binary relation over sets

X

and

Y

, and

S

is a binary relation over sets

Y

and

Z

then

S\circR=\{(x,z)\midthereexistsy\inYsuchthatxRyandySz\}

(also denoted by

R;S

) is the of

R

and

S

over

X

and

Z

.

The identity element is the identity relation. The order of

R

and

S

in the notation

S\circR,

used here agrees with the standard notational order for composition of functions. For example, the composition (is parent of)

\circ

(is mother of) yields (is maternal grandparent of), while the composition (is mother of)

\circ

(is parent of) yields (is grandmother of). For the former case, if

x

is the parent of

y

and

y

is the mother of

z

, then

x

is the maternal grandparent of

z

.

Converse

See main article: Converse relation.

See also: Duality (order theory). If

R

is a binary relation over sets

X

and

Y

then

Rsf{T}=\{(y,x)\midxRy\}

is the,[14] also called,[15] of

R

over

Y

and

X

.

For example,

=

is the converse of itself, as is

≠

and

<

and

>

are each other's converse, as are

\leq

and

\geq

. A binary relation is equal to its converse if and only if it is symmetric.

Complement

If

R

is a binary relation over sets

X

and

Y

then

\bar{R}=\{(x,y)\mid\negxRy\}

(also denoted by

\negR

) is the of

R

over

X

and

Y

.

For example,

=

and

≠

are each other's complement, as are

\subseteq

and

\not\subseteq

,

\supseteq

and

\not\supseteq

,

\in

and

\not\in

, and for total orders also

<

and

\geq

, and

>

and

\leq

.

Rsf{T}

is the converse of the complement:

\overline{RT

} = \bar^\mathsf.

If

X=Y,

the complement has the following properties:

Restriction

See main article: Restriction (mathematics). If

R

is a binary homogeneous relation over a set

X

and

S

is a subset of

X

then

R\vert=\{(x,y)\midxRyandx\inSandy\inS\}

is the of

R

to

S

over

X

.

If

R

is a binary relation over sets

X

and

Y

and if

S

is a subset of

X

then

R\vert=\{(x,y)\midxRyandx\inS\}

is the of

R

to

S

over

X

and

Y

.

If

R

is a binary relation over sets

X

and

Y

and if

S

is a subset of

Y

then

R\vert=\{(x,y)\midxRyandy\inS\}

is the of

R

to

S

over

X

and

Y

.

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, then so too are its restrictions.

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 does not 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.

Also, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers a property of the relation

\leq

is that every non-empty subset

S\subseteq\R

with an upper bound in

\R

has a least upper bound (also called supremum) in

\R.

However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation

\leq

to the rational numbers.

A binary relation

R

over sets

X

and

Y

is said to be a relation

S

over

X

and

Y

, written

R\subseteqS,

if

R

is a subset of

S

, that is, for all

x\inX

and

y\inY,

if

xRy

, then

xSy

. If

R

is contained in

S

and

S

is contained in

R

, then

R

and

S

are called written

R=S

. If

R

is contained in

S

but

S

is not contained in

R

, then

R

is said to be than

S

, written

R\subsetneqS.

For example, on the rational numbers, the relation

>

is smaller than

\geq

, and equal to the composition

>\circ>

.

Matrix representation

Binary relations over sets

X

and

Y

can be represented algebraically by logical matrices indexed by

X

and

Y

with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over

X

and

Y

and a relation over

Y

and

Z

),[16] the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. Homogeneous relations (when

X=Y

) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.[17]

Examples

ball! scope="col"
cardollcup
John + − − −
Mary − − + −
Venus − + − −
ball! scope="col"
cardollcup
John + − − −
Mary − − + −
Ian − − − −
Venus − + − −

Types of binary relations

Some important types of binary relations

R

over sets

X

and

Y

are listed below.

Uniqueness properties:

x,y\inX

and all

z\inY,

if

xRz

and

yRz

then

x=y

. In other words, every element of the codomain has at most one preimage element. For such a relation,

Y

is called a primary key of

R

. For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both

-1

and

1

to

1

), nor the black one (as it relates both

-1

and

1

to

0

).

x\inX

and all

y,z\inY,

if

xRy

and

xRz

then

y=z

. In other words, every element of the domain has at most one image element. Such a binary relation is called a or .[23] For such a relation,

\{X\}

is called of

R

. For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates

1

to both

1

and

-1

), nor the black one (as it relates

0

to both

-1

and

1

).

Totality properties (only definable if the domain

X

and codomain

Y

are specified):

x\inX

there exists a

y\inY

such that

xRy

. In other words, every element of the domain has at least one image element. In other words, the domain of definition of

R

is equal to

X

. This property, is different from the definition of (also called by some authors) in Properties. Such a binary relation is called a . For example, the red and green binary relations in the diagram are total, but the blue one is not (as it does not relate

-1

to any real number), nor the black one (as it does not relate

2

to any real number). As another example,

>

is a total relation over the integers. But it is not a total relation over the positive integers, because there is no

y

in the positive integers such that

1>y

.[24] However,

<

is a total relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is total: for a given

x

, choose

y=x

.

y\inY

, there exists an

x\inX

such that

xRy

. In other words, every element of the codomain has at least one preimage element. In other words, the codomain of definition of

R

is equal to

Y

. For example, the green and blue binary relations in the diagram are surjective, but the red one is not (as it does not relate any real number to

-1

), nor the black one (as it does not relate any real number to

2

).

Uniqueness and totality properties (only definable if the domain

X

and codomain

Y

are specified):

If relations over proper classes are allowed:

x\inX

, the class of all

y\inY

such that

yRx

, i.e.

\{y\inY,yRx\}

, is a set. For example, the relation

\in

is set-like, and every relation on two sets is set-like.[25] The usual ordering < over the class of ordinal numbers is a set-like relation, while its inverse > is not.

Sets versus classes

Certain mathematical "relations", such as "equal to", "subset of", and "member 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, to model the general concept of "equality" as a binary relation

=

, 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

\subseteq

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 by

\subseteqA.

Also, the "member of" relation needs to be restricted to have domain

A

and codomain

P(A)

to obtain a binary relation

\inA

that is a set. Bertrand Russell has shown that assuming

\in

to be defined over all sets leads to a contradiction in naive set theory, see Russell's paradox.

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 binary relation with its graph in this context.)[26] With this definition one can for instance define a binary relation over every set and its power set.

Homogeneous relation

See main article: Homogeneous relation. A homogeneous relation over a set

X

is a binary relation over

X

and itself, i.e. it is a subset of the Cartesian product

X x X.

[12] [27] [28] It is also simply called a (binary) relation over

X

.

A homogeneous relation

R

over a set

X

may be identified with a directed simple graph permitting loops, where

X

is the vertex set and

R

is the edge set (there is an edge from a vertex

x

to a vertex

y

if and only if

xRy

).The set of all homogeneous relations

l{B}(X)

over a set

X

is the power set

2X

which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on

l{B}(X)

, it forms a semigroup with involution.

Some important properties that a homogeneous relation

R

over a set

X

may have are:

x\inX,

xRx

. For example,

\geq

is a reflexive relation but > is not.

x\inX,

not

xRx

. For example,

>

is an irreflexive relation, but

\geq

is not.

x,y\inX,

if

xRy

then

yRx

. For example, "is a blood relative of" is a symmetric relation.

x,y\inX,

if

xRy

and

yRx

then

x=y.

For example,

\geq

is an antisymmetric relation.

x,y\inX,

if

xRy

then not

yRx

. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[29] For example, > is an asymmetric relation, but

\geq

is not.

x,y,z\inX,

if

xRy

and

yRz

then

xRz

. A transitive relation is irreflexive if and only if it is asymmetric.[30] For example, "is ancestor of" is a transitive relation, while "is parent of" is not.

x,y\inX,

if

x ≠ y

then

xRy

or

yRx

.

x,y\inX,

xRy

or

yRx

.

x,y\inX,

if

xRy,

then some

z\inX

exists such that

xRz

and

zRy

.

A is a relation that is reflexive, antisymmetric, and transitive. A is a relation that is irreflexive, asymmetric, and transitive. A is a relation that is reflexive, antisymmetric, transitive and connected.[31] A is a relation that is irreflexive, asymmetric, transitive and connected.An is a relation that is reflexive, symmetric, and transitive.For example, "

x

divides

y

" is a partial, but not a total order on natural numbers

\N,

"

x<y

" is a strict total order on

\N,

and "

x

is parallel to

y

" is an equivalence relation on the set of all lines in the Euclidean plane.

All operations defined in section also apply to homogeneous relations.Beyond that, a homogeneous relation over a set

X

may be subjected to closure operations like:
: the smallest reflexive relation over

X

containing

R

,
: the smallest transitive relation over

X

containing

R

,
: the smallest equivalence relation over

X

containing

R

.

Calculus of relations

Developments in algebraic logic have facilitated usage of binary relations. The calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations. The inclusion

R\subseteqS,

meaning that

aRb

implies

aSb

, sets the scene in a lattice of relations. But since

P\subseteqQ\equiv(P\cap\bar{Q}=\varnothing)\equiv(P\capQ=P),

the inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of

A x B.

In contrast to homogeneous relations, the composition of relations operation is only a partial function. The necessity of matching target to source of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations. The of the category Rel are sets, and the relation-morphisms compose as required in a category.

Induced concept lattice

Binary relations have been described through their induced concept lattices:A concept

C\subsetR

satisfies two properties:

C

is the outer product of logical vectors

Ci=uivj,   u,v

logical vectors.

C

is maximal, not contained in any other outer product. Thus

C

is described as a non-enlargeable rectangle.

For a given relation

R\subseteqX x Y,

the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion

\sqsubseteq

forming a preorder.

The MacNeille completion theorem (1937) (that any partial order may be embedded in a complete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices".[32] The decomposition is

R=fEgsf{T}

, where

f

and

g

are functions, called or left-total, functional relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial order

E

that belongs to the minimal decomposition

(f,g,E)

of the relation

R

."

Particular cases are considered below:

E

total order corresponds to Ferrers type, and

E

identity corresponds to difunctional, a generalization of equivalence relation on a set.

Relations may be ranked by the Schein rank which counts the number of concepts necessary to cover a relation.[33] Structural analysis of relations with concepts provides an approach for data mining.[34]

Particular relations

R

is a serial relation and

RT

is its transpose, then

I\subseteqRsf{T}R

where

I

is the

m x m

identity relation.

R

is a surjective relation, then

I\subseteqRRsf{T}

where

I

is the

n x n

identity relation.

Difunctional

The idea of a difunctional relation is to partition objects by distinguishing attributes, as a generalization of the concept of an equivalence relation. One way this can be done is with an intervening set

Z=\{x,y,z,\ldots\}

of indicators. The partitioning relation

R=FGsf{T}

is a composition of relations using relations

F\subseteqA x ZandG\subseteqB x Z.

Jacques Riguet named these relations difunctional since the composition

FGT

involves functional relations, commonly called partial functions.

In 1950 Riguet showed that such relations satisfy the inclusion:[35]

R R^\textsf R \subseteq R

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This terminology recalls the fact that, when represented as a logical matrix, the columns and rows of a difunctional relation can be arranged as a block matrix with rectangular blocks of ones on the (asymmetric) main diagonal.[36] More formally, a relation

R

on

X x Y

is difunctional if and only if it can be written as the union of Cartesian products

Ai x Bi

, where the

Ai

are a partition of a subset of

X

and the

Bi

likewise a partition of a subset of

Y

.[37]

Using the notation

\{y\midxRy\}=xR

, a difunctional relation can also be characterized as a relation

R

such that wherever

x1R

and

x2R

have a non-empty intersection, then these two sets coincide; formally

x1\capx2 ≠ \varnothing

implies

x1R=x2R.

[38]

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management." Furthermore, difunctional relations are fundamental in the study of bisimulations.[39]

In the context of homogeneous relations, a partial equivalence relation is difunctional.

Ferrers type

A strict order on a set is a homogeneous relation arising in order theory.In 1951 Jacques Riguet adopted the ordering of an integer partition, called a Ferrers diagram, to extend ordering to binary relations in general.[40]

The corresponding logical matrix of a general binary relation has rows which finish with a sequence of ones. Thus the dots of a Ferrer's diagram are changed to ones and aligned on the right in the matrix.

An algebraic statement required for a Ferrers type relation R isR \bar^\textsf R \subseteq R.

If any one of the relations

R,\bar{R},Rsf{T}

is of Ferrers type, then all of them are.[41]

Contact

Suppose

B

is the power set of

A

, the set of all subsets of

A

. Then a relation

g

is a contact relation if it satisfies three properties:

forallx\inA,Y=\{x\}impliesxgY.

Y\subseteqZandxgYimpliesxgZ.

forally\inY,ygZandxgYimpliesxgZ.

The set membership relation,

\epsilon=

"is an element of", satisfies these properties so

\epsilon

is a contact relation. The notion of a general contact relation was introduced by Georg Aumann in 1970.[42] [43]

In terms of the calculus of relations, sufficient conditions for a contact relation includeC^\textsf \bar \subseteq \ni \bar \equiv C \overline \subseteq C, where

\ni

is the converse of set membership (

\in

).

Preorder R\R

Every relation

R

generates a preorder

R\backslashR

which is the left residual.[44] In terms of converse and complements,

R\backslashR\equiv\overline{Rsf{T}\bar{R}}.

Forming the diagonal of

Rsf{T}\bar{R}

, the corresponding row of

Rsf{T

} and column of

\bar{R}

will be of opposite logical values, so the diagonal is all zeros. Then

Rsf{T}\bar{R}\subseteq\bar{I}\impliesI\subseteq\overline{Rsf{T}\bar{R}}=R\backslashR

, so that

R\backslashR

is a reflexive relation.

To show transitivity, one requires that

(R\backslashR)(R\backslashR)\subseteqR\backslashR.

Recall that

X=R\backslashR

is the largest relation such that

RX\subseteqR.

Then

R(R\backslashR)\subseteqR

R(R\backslashR)(R\backslashR)\subseteqR

(repeat)

\equivRsf{T}\bar{R}\subseteq\overline{(R\backslashR)(R\backslashR)}

(Schröder's rule)

\equiv(R\backslashR)(R\backslashR)\subseteq\overline{Rsf{T}\bar{R}}

(complementation)

\equiv(R\backslashR)(R\backslashR)\subseteqR\backslashR.

(definition)

The inclusion relation Ω on the power set of

U

can be obtained in this way from the membership relation

\in

on subsets of

U

:

\Omega=\overline{\ni\bar{\in}}=\in\backslash\in.

Fringe of a relation

Given a relation

R

, its fringe is the sub-relation defined as\operatorname(R) = R \cap \overline.

When

R

is a partial identity relation, difunctional, or a block diagonal relation, then

\operatorname{fringe}(R)=R

. Otherwise the

\operatorname{fringe}

operator selects a boundary sub-relation described in terms of its logical matrix:

\operatorname{fringe}(R)

is the side diagonal if

R

is an upper right triangular linear order or strict order.

\operatorname{fringe}(R)

is the block fringe if

R

is irreflexive (

R\subseteq\bar{I}

) or upper right block triangular.

\operatorname{fringe}(R)

is a sequence of boundary rectangles when

R

is of Ferrers type.

On the other hand,

\operatorname{fringe}(R)=\emptyset

when

R

is a dense, linear, strict order.[45]

Mathematical heaps

See main article: Heap (mathematics). Given two sets

A

and

B

, the set of binary relations between them

l{B}(A,B)

can be equipped with a ternary operation

[a,b,c]=absf{T}c

where

bT

denotes the converse relation of

b

. In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps, heaps, and generalized heaps.[46] The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:

See also

Bibliography

Notes and References

  1. Web site: Meyer. Albert. 17 November 2021. MIT 6.042J Math for Computer Science, Lecture 3T, Slide 2. live. https://web.archive.org/web/20211117161447/https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2015/lecture-slides/MIT6_042JS16_Relations.pdf . 2021-11-17 .
  2. Codd . Edgar Frank . Edgar F. Codd. June 1970 . A Relational Model of Data for Large Shared Data Banks . https://web.archive.org/web/20040908011134/http://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf . 2004-09-08 . live . Communications of the ACM . 13 . 6 . 377–387 . 10.1145/362384.362685 . 207549016 . 2020-04-29.
  3. Web site: Relation definition – Math Insight. mathinsight.org. 2019-12-11.
  4. [Ernst Schröder (mathematician)|Ernst Schröder]
  5. [C. I. Lewis]
  6. [Gunther Schmidt]
  7. Book: Suppes, Patrick . Patrick Suppes . 1972 . Axiomatic Set Theory . Dover . originally published by D. van Nostrand Company in 1960 . 0-486-61630-4 . registration .
  8. Book: Smullyan . Raymond M. . Raymond Smullyan . Fitting . Melvin . 2010 . Set Theory and the Continuum Problem . Dover . revised and corrected republication of the work originally published in 1996 by Oxford University Press, New York . 978-0-486-47484-7.
  9. Book: Levy, Azriel . Azriel Levy . 2002 . Basic Set Theory . Dover . republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979 . 0-486-42079-5.
  10. Book: Schmidt. Gunther. Ströhlein. Thomas. [{{google books |plainurl=y |id=ZgarCAAAQBAJ|paged=277}} Relations and Graphs: Discrete Mathematics for Computer Scientists]. 2012. Springer Science & Business Media. 978-3-642-77968-8. Gunther Schmidt . Definition 4.1.1..
  11. Book: Christodoulos A. Floudas. Christodoulos Floudas. Panos M. Pardalos. Encyclopedia of Optimization. 2008. Springer Science & Business Media. 978-0-387-74758-3. 299–300. 2nd.
  12. Book: Michael Winter. Goguen Categories: A Categorical Approach to L-fuzzy Relations. 2007. Springer. 978-1-4020-6164-6. x-xi.
  13. Jacobson, Nathan (2009), Basic Algebra II (2nd ed.) § 2.1.
  14. [Garrett Birkhoff]
  15. [Mary P. Dolciani]
  16. quantum mechanics over a commutative rig . John C. Baez . John C. Baez . 6 Nov 2001 . sci.physics.research . 9s87n0$iv5@gap.cco.caltech.edu . November 25, 2018.
  17. Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28., pp. 7-10
  18. Van Gasteren 1990, p. 45.
  19. Kilp, Knauer, Mikhalev 2000, p. 3.
  20. Web site: Functional relation - Encyclopedia of Mathematics. 2024-06-13. encyclopediaofmath.org.
  21. Web site: functional relation in nLab. 2024-06-13. ncatlab.org.
  22. Schmidt 2010, p. 49.
  23. Kilp, Knauer, Mikhalev 2000, p. 4.
  24. Yao. Y.Y.. Wong, S.K.M.. Generalization of rough sets using relationships between attribute values. Proceedings of the 2nd Annual Joint Conference on Information Sciences. 1995. 30–33. .
  25. Book: Set theory: an introduction to independence proofs. 102 . registration. Kunen . Kenneth. North-Holland. 1980. 0-444-85401-0. 0443.03021.
  26. Book: A formalization of set theory without variables . Tarski . Alfred . Alfred Tarski . Givant . Steven . 1987 . 3 . American Mathematical Society . 0-8218-1041-3 .
  27. Book: M. E. Müller. Relational Knowledge Discovery. 2012. Cambridge University Press. 978-0-521-19021-3. 22.
  28. Book: Peter J. Pahl. Rudolf Damrath. Mathematical Foundations of Computational Engineering: A Handbook. 2001. Springer Science & Business Media. 978-3-540-67995-0. 496.
  29. .
  30. Book: Flaška. V.. Ježek. J.. Kepka. T.. Kortelainen. J.. Transitive Closures of Binary Relations I. 2007. School of Mathematics – Physics Charles University. Prague. 1. dead. https://web.archive.org/web/20131102214049/http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf. 2013-11-02. Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
  31. Joseph G. Rosenstein, Linear orderings, Academic Press, 1982,, p. 4
  32. [R. Berghammer]
  33. [Ki-Hang Kim]
  34. Ali Jaoua, Rehab Duwairi, Samir Elloumi, and Sadok Ben Yahia (2009) "Data mining, reasoning and incremental information retrieval through non enlargeable rectangular relation coverage", pages 199 to 210 in Relations and Kleene algebras in computer science, Lecture Notes in Computer Science 5827, Springer
  35. Riguet . Jacques. Jacques Riguet. Comptes rendus . January 1950 . Quelques proprietes des relations difonctionelles. fr. 230. 1999–2000.
  36. Book: Julius Richard Büchi. Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions. 1989. Springer Science & Business Media. 978-1-4613-8853-1. 35–37. Julius Richard Büchi.
  37. East . James . Vernitski . Alexei . Ranks of ideals in inverse semigroups of difunctional binary relations . Semigroup Forum . February 2018 . 96 . 1 . 21–30 . 10.1007/s00233-017-9846-9. 1612.04935. 54527913 .
  38. Book: Chris Brink. Wolfram Kahl. Gunther Schmidt. Relational Methods in Computer Science. 1997. Springer Science & Business Media. 978-3-211-82971-4. 200.
  39. Book: 10.1007/978-3-662-44124-4_7. Coalgebraic Simulations and Congruences. Coalgebraic Methods in Computer Science. 8446. 118. Lecture Notes in Computer Science. 2014. Gumm . H. P. . Zarrad . M. . 978-3-662-44123-7.
  40. J. Riguet (1951) "Les relations de Ferrers", Comptes Rendus 232: 1729,30
  41. Book: Schmidt. Gunther. Ströhlein. Thomas. [{{google books |plainurl=y |id=ZgarCAAAQBAJ|paged=277}} Relations and Graphs: Discrete Mathematics for Computer Scientists]. 2012. Springer Science & Business Media. 978-3-642-77968-8. Gunther Schmidt . 77.
  42. Kontakt-Relationen . Georg Aumann . Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften München . 1970 . II . 67 - 77 . 1971 .
  43. Anne K. Steiner (1970) Review:Kontakt-Relationen from Mathematical Reviews
  44. In this context, the symbol

    \backslash

    does not mean "set difference".
  45. [Gunther Schmidt]
  46. [Viktor Wagner]

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