Английская Википедия:Binary relation

Шаблон:Short description Шаблон:Hatnote Шаблон:Binary relations In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain.^[1] A binary relation over sets Шаблон:Mvar and Шаблон:Mvar is a new set of ordered pairs Шаблон:Math consisting of elements Шаблон:Mvar from Шаблон:Mvar and Шаблон:Mvar from Шаблон:Mvar.^[2] It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element Шаблон:Mvar is related to an element Шаблон:Mvar, if and only if the pair Шаблон:Math belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case Шаблон:Math of an [[Finitary relation|Шаблон:Mvar-ary relation]] over sets Шаблон:Math, which is a subset of the Cartesian product <math>X_1 \times \cdots \times X_n.</math>^[2]

An example of a binary relation is the "divides" relation over the set of prime numbers <math>\mathbb{P}</math> and the set of integers <math>\mathbb{Z}</math>, in which each prime Шаблон:Mvar is related to each integer Шаблон:Mvar that is a multiple of Шаблон:Mvar, but not to an integer that is not a multiple of Шаблон:Mvar. 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:

the "is greater than", "is equal to", and "divides" relations in arithmetic;
the "is congruent to" relation in geometry;
the "is adjacent to" relation in graph theory;
the "is orthogonal to" relation in linear algebra.

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 Шаблон:Mvar and Шаблон:Mvar is an element of the power set of <math>X \times Y.</math> Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of <math>X \times Y.</math> 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 Шаблон:Em, 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.

The terms Шаблон:Em,^[7] 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 <math>X \times Y</math> without reference to Шаблон:Mvar and Шаблон:Mvar, and reserve the term "correspondence" for a binary relation with reference to Шаблон:Mvar and Шаблон:Mvar.Шаблон:Citation needed

Definition

Given sets X and Y, the Cartesian product <math>X \times Y</math> is defined as <math>\{ (x, y) : x \in X \text{ and } y \in Y \},</math> and its elements are called ordered pairs.

A Шаблон:Em R over sets X and Y is a subset of <math>X \times Y.</math>^[2]^[8] The set X is called the Шаблон:Em^[2] or Шаблон:Em of R, and the set Y the Шаблон:Em or Шаблон:Em of R. In order to specify the choices of the sets X and Y, some authors define a Шаблон:Em or Шаблон:Em as an ordered triple Шаблон:Math, where G is a subset of <math>X \times Y</math> called the Шаблон:Em of the binary relation. The statement <math>(x, y) \in R</math> reads "x is R-related to y" and is denoted by xRy.^[4]^[5]^[6]^{[note 1]} The Шаблон:Em or Шаблон:Em^[2] of R is the set of all x such that xRy for at least one y. The codomain of definition, Шаблон:Em,^[2] Шаблон:Em or Шаблон:Em of R is the set of all y such that xRy for at least one x. The Шаблон:Em of R is the union of its domain of definition and its codomain of definition.^[10]^[11]^[12]

When <math>X = Y,</math> a binary relation is called a Шаблон:Em (or Шаблон:Em). To emphasize the fact that X and Y are allowed to be different, a binary relation is also called a heterogeneous relation.^[13]^[14]^[15]

In a binary relation, the order of the elements is important; if <math>x \neq y</math> 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 <math>R \cup S = \{ (x, y) : xRy \text{ or } xSy \}</math> is the Шаблон:Em of R and S over X and Y.

The identity element is the empty relation. For example, <math>\,\leq\,</math> is the union of < and =, and <math>\,\geq\,</math> is the union of > and =.

Intersection

If R and S are binary relations over sets X and Y then <math>R \cap S = \{ (x, y) : xRy \text{ and } xSy \}</math> is the Шаблон:Em 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

Шаблон:Main If R is a binary relation over sets X and Y, and S is a binary relation over sets Y and Z then <math>S \circ R = \{ (x, z) : \text{ there exists } y \in Y \text{ such that } xRy \text{ and } ySz \}</math> (also denoted by Шаблон:Math) is the Шаблон:Em of R and S over X and Z.

The identity element is the identity relation. The order of R and S in the notation <math>S \circ R,</math> used here agrees with the standard notational order for composition of functions. For example, the composition (is parent of)<math>\,\circ\,</math>(is mother of) yields (is maternal grandparent of), while the composition (is mother of)<math>\,\circ\,</math>(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

Шаблон:Main Шаблон:See also If R is a binary relation over sets X and Y then <math>R^\textsf{T} = \{ (y, x) : xRy \}</math> is the Шаблон:Em,Шаблон:Cn also called Шаблон:Em,Шаблон:Cn of R over Y and X.

For example, <math>\,=\,</math> is the converse of itself, as is <math>\,\neq,\,</math> and <math>\,<\,</math> and <math>\,>\,</math> are each other's converse, as are <math>\,\leq\,</math> and <math>\,\geq.\,</math> A binary relation is equal to its converse if and only if it is symmetric.

Complement

Шаблон:Main If R is a binary relation over sets X and Y then <math>\overline{R} = \{ (x, y) : \text{ not } xRy \}</math> (also denoted by Шаблон:Strikethrough or Шаблон:Math) is the Шаблон:Em of R over X and Y.

For example, <math>\,=\,</math> and <math>\,\neq\,</math> are each other's complement, as are <math>\,\subseteq\,</math> and <math>\,\not\subseteq,\,</math> <math>\,\supseteq\,</math> and <math>\,\not\supseteq,\,</math> and <math>\,\in\,</math> and <math>\,\not\in,\,</math> and, for total orders, also <math>\,<\,</math> and <math>\,\geq,\,</math> and <math>\,>\,</math> and <math>\,\leq.\,</math>

The complement of the converse relation <math>R^\textsf{T}</math> is the converse of the complement: <math>\overline{R^\mathsf{T}} = \bar{R}^\mathsf{T}.</math>

If <math>X = Y,</math> the complement has the following properties:

If a relation is symmetric, then so is the complement.
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.

Restriction

Шаблон:Main If R is a binary homogeneous relation over a set X and S is a subset of X then <math>R_{\vert S} = \{ (x, y) \mid xRy \text{ and } x \in S \text{ and } y \in S \}</math> is the Шаблон:Em 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 <math>R_{\vert S} = \{ (x, y) \mid xRy \text{ and } x \in S \}</math> is the Шаблон:Em of R to S over X and Y.Шаблон:Clarify

If R is a binary relation over sets X and Y and if S is a subset of Y then <math>R^{\vert S} = \{ (x, y) \mid xRy \text{ and } y \in S \}</math> is the Шаблон:Em 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 <math>\,\leq\,</math> is that every non-empty subset <math>S \subseteq \R</math> with an upper bound in <math>\R</math> has a least upper bound (also called supremum) in <math>\R.</math> However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation <math>\,\leq\,</math> to the rational numbers.

A binary relation R over sets X and Y is said to be Шаблон:Em a relation S over X and Y, written <math>R \subseteq S,</math> if R is a subset of S, that is, for all <math>x \in X</math> and <math>y \in Y,</math> if xRy, then xSy. If R is contained in S and S is contained in R, then R and S are called Шаблон:Em written R = S. If R is contained in S but S is not contained in R, then R is said to be Шаблон:Em than S, written <math>R \subsetneq S.</math> For example, on the rational numbers, the relation <math>\,>\,</math> is smaller than <math>\,\geq,\,</math> and equal to the composition <math>\,>\,\circ\,>.\,</math>

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

Examples

2nd example relation
Шаблон:Diagonal split header	ball	car	doll	cup
John	+	−	−	−
Mary	−	−	+	−
Venus	−	+	−	−

1st example relation
Шаблон:Diagonal split header	ball	car	doll	cup
John	+	−	−	−
Mary	−	−	+	−
Ian	−	−	−	−
Venus	−	+	−	−

Шаблон:Olist

Specific types of binary relations

Файл:The four types of binary relations.png

Examples of four types of binary relations over the real numbers: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black).

Some important types of binary relations R over sets X and Y are listed below.

Uniqueness properties:

Injective (also called left-unique):^[18] for all <math>x, z \in X</math> and all <math>y \in Y,</math> if Шаблон:Math and Шаблон:Math then Шаблон:Math. For such a relation, Шаблон:Mset is called a primary key of R.^[2] 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).
Univalent^[6] (also called right-unique,^[18] right-definite^[19] or functional^[20]): for all <math>x \in X</math> and all <math>y, z \in Y,</math> if Шаблон:Math and Шаблон:Math then Шаблон:Math. Such a binary relation is called a Шаблон:Em. For such a relation, <math>\{ X \}</math> is called Шаблон:Em of R.^[2] For example, the red and green binary relations in the diagram are univalent, 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).
One-to-one: injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.
One-to-many: injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.
Many-to-one: functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.
Many-to-many: not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not.

Totality properties (only definable if the domain X and codomain Y are specified):

Total (also called left-total):^[18] for all x in X there exists a y in Y such that Шаблон:Math. In other words, the domain of definition of R is equal to X. This property, is different from the definition of Шаблон:Em (also called Шаблон:Em by some authors)Шаблон:Citation needed in Properties. Such a binary relation is called a Шаблон:Em. 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 Шаблон:Mvar in the positive integers such that Шаблон:Math.^[21] However, < is a total relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is total: for a given Шаблон:Mvar, choose Шаблон:Math.
Surjective (also called right-total^[18] or onto): for all y in Y, there exists an x in X such that xRy. 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):

A Шаблон:Em: a binary relation that is functional and total. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
An Шаблон:Em: a function that is injective. For example, the green binary relation in the diagram is an injection, but the red, blue and black ones are not.
A Шаблон:Em: a function that is surjective. For example, the green binary relation in the diagram is a surjection, but the red, blue and black ones are not.
A Шаблон:Em: a function that is injective and surjective. For example, the green binary relation in the diagram is a bijection, but the red, blue and black ones are not.

Шаблон:AnchorIf relations over proper classes are allowed:

Set-like (or Шаблон:Em): for all Шаблон:Mvar in Шаблон:Mvar, the class of all Шаблон:Mvar in Шаблон:Mvar such that Шаблон:Math, i.e. <math>\{y\in Y : yRx\}</math>, is a set. For example, the relation <math>\in</math> is set-like, and every relation on two sets is set-like.^[22] The usual ordering < over the class of ordinal numbers is a set-like relation, while its inverse > is not.Шаблон:Citation needed

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 <math>\,=,</math> 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 <math>\,\subseteq\,</math> 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 <math>\,\subseteq_A.\,</math> Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation <math>\,\in_A\,</math> that is a set. Bertrand Russell has shown that assuming <math>\,\in\,</math> 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 Шаблон:Math, 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.)^[23] With this definition one can for instance define a binary relation over every set and its power set.

Homogeneous relation

Шаблон:Main 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 <math>X \times X.</math>^[15]^[24]^[25] 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 Шаблон:Math). The set of all homogeneous relations <math>\mathcal{B}(X)</math> over a set X is the power set <math>2^{X \times X}</math> 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 <math>\mathcal{B}(X)</math>, it forms a semigroup with involution.

Some important properties that a homogeneous relation Шаблон:Mvar over a set Шаблон:Mvar may have are:

Шаблон:Em: for all <math>x \in X,</math> Шаблон:Math. For example, <math>\,\geq\,</math> is a reflexive relation but > is not.
Шаблон:Em: for all <math>x \in X,</math> not Шаблон:Math. For example, <math>\,>\,</math> is an irreflexive relation, but <math>\,\geq\,</math> is not.
Шаблон:Em: for all <math>x, y \in X,</math> if Шаблон:Math then Шаблон:Math. For example, "is a blood relative of" is a symmetric relation.
Шаблон:Em: for all <math>x, y \in X,</math> if Шаблон:Math and Шаблон:Math then <math>x = y.</math> For example, <math>\,\geq\,</math> is an antisymmetric relation.^[26]
Шаблон:Em: for all <math>x, y \in X,</math> if Шаблон:Math then not Шаблон:Math. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.^[27] For example, > is an asymmetric relation, but <math>\,\geq\,</math> is not.
Шаблон:Em: for all <math>x, y, z \in X,</math> if Шаблон:Math and Шаблон:Math then Шаблон:Math. A transitive relation is irreflexive if and only if it is asymmetric.^[28] For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
Шаблон:Em: for all <math>x, y \in X,</math> if <math>x \neq y</math> then Шаблон:Math or Шаблон:Math.
Шаблон:Em: for all <math>x, y \in X,</math> Шаблон:Math or Шаблон:Math.
Шаблон:Em: for all <math>x, y \in X,</math> if <math>xRy ,</math> then some <math>z \in X</math> exists such that <math>xRz</math> and <math>zRy</math>.

A Шаблон:Em is a relation that is reflexive, antisymmetric, and transitive. A Шаблон:Em is a relation that is irreflexive, asymmetric, and transitive. A Шаблон:Em is a relation that is reflexive, antisymmetric, transitive and connected.^[29] A Шаблон:Em is a relation that is irreflexive, asymmetric, transitive and connected. An Шаблон:Em 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 <math>\N,</math> "x < y" is a strict total order on <math>\N,</math> 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 Шаблон:Slink also apply to homogeneous relations. Beyond that, a homogeneous relation over a set X may be subjected to closure operations like:

Шаблон:Em: the smallest reflexive relation over X containing R,
Шаблон:Em: the smallest transitive relation over X containing R,
Шаблон:Em: the smallest equivalence relation over X containing R.

Heterogeneous relation

In mathematics, a heterogeneous relation is a binary relation, a subset of a Cartesian product <math>A \times B,</math> where A and B are possibly distinct sets.^[30] The prefix hetero is from the Greek ἕτερος (heteros, "other, another, different").

A heterogeneous relation has been called a rectangular relation,^[15] suggesting that it does not have the square-like symmetry of a homogeneous relation on a set where <math>A = B.</math> 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 Шаблон:Em or Шаблон:Em, i.e. as relations where the normal case is that they are relations between different sets."^[31]

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 <math>R \subseteq S,</math> meaning that aRb implies aSb, sets the scene in a lattice of relations. But since <math>P \subseteq Q \equiv (P \cap \bar{Q} = \varnothing ) \equiv (P \cap Q = P),</math> 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 <math>A \times B.</math>

In contrast to homogeneous relations, the composition of relations operation is only a partial function. The necessity of matching range to domain 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 Шаблон:Em of the category Rel are sets, and the relation-morphisms compose as required in a category.Шаблон:Citation needed

Induced concept lattice

Binary relations have been described through their induced concept lattices: A concept C ⊂ R satisfies two properties: (1) The logical matrix of C is the outer product of logical vectors

<math>C_{i j} \ = \ u_i v_j , \quad u, v</math> logical vectors.Шаблон:Clarify (2) C is maximal, not contained in any other outer product. Thus C is described as a non-enlargeable rectangle.

For a given relation <math>R \subseteq X \times Y,</math> the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion <math>\sqsubseteq</math> 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

<math>R \ = \ f \ E \ g^\textsf{T} ,</math> where f and g are functions, called Шаблон:Em or left-total, univalent 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

Proposition: If R is a serial relation and R^T is its transpose, then <math>I \subseteq R^\textsf{T} R</math> where Шаблон:Mvar is the m × m identity relation.
Proposition: If R is a surjective relation, then <math>I \subseteq R R^\textsf{T}</math> where Шаблон:Mvar is the <math>n \times n</math> identity relation.

Difunctional

Шаблон:Anchor 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 <math>Z = \{ x, y, z, \ldots \}</math> of indicators. The partitioning relation <math>R = F G^\textsf{T}</math> is a composition of relations using Шаблон:Em relations <math>F \subseteq A \times Z \text{ and } G \subseteq B \times Z.</math> Jacques Riguet named these relations difunctional since the composition F G^T involves univalent relations, commonly called partial functions.

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

<math display=block>R \ R^\textsf{T} \ R \ \subseteq \ R</math>

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 Шаблон:Mvar on <math>X \times Y</math> is difunctional if and only if it can be written as the union of Cartesian products <math>A_i \times B_i</math>, where the <math>A_i</math> are a partition of a subset of Шаблон:Mvar and the <math>B_i</math> likewise a partition of a subset of Шаблон:Mvar.^[37]

Using the notation {y: xRy} = 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 <math>x_1 \cap x_2 \neq \varnothing</math> implies <math>x_1 R = x_2 R.</math>^[38]

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management."^[39] Furthermore, difunctional relations are fundamental in the study of bisimulations.^[40]

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 a partition of an integer, called a Ferrers diagram, to extend ordering to binary relations in general.^[41]

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 is <math display="block">R \bar{R}^\textsf{T} R \subseteq R.</math>

If any one of the relations <math>R, \ \bar{R}, \ R^\textsf{T}</math> is of Ferrers type, then all of them are. ^[30]

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:

<math>\text{for all } x \in A, Y = \{ x \} \text{ implies } xgY.</math>
<math>Y \subseteq Z \text{ and } xgY \text{ implies } xgZ.</math>
<math>\text{for all } y \in Y, ygZ \text{ and } xgY \text{ implies } xgZ.</math>

The set membership relation, ε = "is an element of", satisfies these properties so ε 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 include <math display="block">C^\textsf{T} \bar{C} \ \subseteq \ \ni \bar{C} \ \ \equiv \ C \ \overline{\ni \bar{C}} \ \subseteq \ C,</math> where <math>\ni</math> is the converse of set membership (Шаблон:Math).^[44]Шаблон:Rp

Preorder R\R

Every relation R generates a preorder <math>R \backslash R</math> which is the left residual.^[45] In terms of converse and complements, <math>R \backslash R \ \equiv \ \overline{R^\textsf{T} \bar{R}}.</math> Forming the diagonal of <math>R^\textsf{T} \bar{R}</math>, the corresponding row of <math>R^{\text{T}}</math> and column of <math>\bar{R}</math> will be of opposite logical values, so the diagonal is all zeros. Then

<math>R^\textsf{T} \bar{R} \subseteq \bar{I} \ \implies \ I \subseteq \overline{R^\textsf{T} \bar{R}} \ = \ R \backslash R ,</math> so that <math>R \backslash R</math> is a reflexive relation.

To show transitivity, one requires that <math>(R\backslash R)(R\backslash R) \subseteq R \backslash R.</math> Recall that <math>X = R \backslash R</math> is the largest relation such that <math>R X \subseteq R.</math> Then

<math>R(R\backslash R) \subseteq R</math>

<math>R(R\backslash R) (R\backslash R )\subseteq R</math> (repeat)

<math>\equiv R^\textsf{T} \bar{R} \subseteq \overline{(R \backslash R)(R \backslash R)}</math> (Schröder's rule)

<math>\equiv (R \backslash R)(R \backslash R) \subseteq \overline{R^\textsf{T} \bar{R}}</math> (complementation)

<math>\equiv (R \backslash R)(R \backslash R) \subseteq R \backslash R.</math> (definition)

The inclusion relation Ω on the power set of U can be obtained in this way from the membership relation <math>\,\in\,</math> on subsets of U:

<math>\Omega \ = \ \overline{\ni \bar{\in}} \ = \ \in \backslash \in .</math>^[44]Шаблон:Rp

Fringe of a relation

Given a relation R, a sub-relation called its Шаблон:Em is defined as <math display="block">\operatorname{fringe}(R) = R \cap \overline{R \bar{R}^\textsf{T} R}.</math>

When R is a partial identity relation, difunctional, or a block diagonal relation, then fringe(R) = R. Otherwise the fringe operator selects a boundary sub-relation described in terms of its logical matrix: fringe(R) is the side diagonal if R is an upper right triangular linear order or strict order. Fringe(R) is the block fringe if R is irreflexive (<math>R \subseteq \bar{I}</math>) or upper right block triangular. Fringe(R) is a sequence of boundary rectangles when R is of Ferrers type.

On the other hand, Fringe(R) = ∅ when R is a dense, linear, strict order.^[44]

Mathematical heaps

Шаблон:Main Given two sets A and B, the set of binary relations between them <math>\mathcal{B}(A,B)</math> can be equipped with a ternary operation <math>[a, \ b,\ c] \ = \ a b^\textsf{T} c</math> where b^T denotes the converse relation of b. In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps, heaps, and generalized heaps.^[46]^[47] The contrast of heterogeneous and homogeneous relations is highlighted by these definitions: Шаблон:Blockquote

Notes

Шаблон:Reflist

References