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{{Short description|Relationship between elements of two sets}}{{hatnote|This article covers advanced notions. For basic topics, see Relation (mathematics).}}{{Binary relations}}In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain.WEB, Meyer, Albert, 17 November 2021, MIT 6.042J Math for Computer Science, Lecture 3T, Slide 2,weblink live,weblink 2021-11-17, A binary relation over sets {{mvar|X}} and {{mvar|Y}} is a set of ordered pairs {{math|(x, y)}} consisting of elements {{mvar|x}} from {{mvar|X}} and {{mvar|y}} from {{mvar|Y}}.JOURNAL, Codd, Edgar Frank < and =, and ,geq, is the union of > and =.

Intersection

If R and S are binary relations over sets X and Y then R cap S = { (x, y) : xRy text{ and } xSy } is the {{em|intersection relation}} 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

If R is a binary relation over sets X and Y, and S is a binary relation over sets Y and Z then S circ R = { (x, z) : text{ there exists } y in Y text{ such that } xRy text{ and } ySz } (also denoted by {{math|R; S}}) is the {{em|composition relation}} 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 circ R, 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 also|Duality (order theory)}}If R is a binary relation over sets X and Y then R^textsf{T} = { (y, x) : xRy } is the {{em|converse relation}},Garrett Birkhoff & Thomas Bartee (1970) Modern Applied Algebra, page 35, McGraw-Hill also called {{em|inverse relation}},Mary P. Dolciani (1962) Modern Algebra: Structure and Method, Book 2, page 339, Houghton Mifflin of R over Y and X.For example, ,=, is the converse of itself, as is ,neq,, 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) : text{ not } xRy } (also denoted by {{strikethrough|R}} or {{math|¬ R}}) is the {{em|complementary relation}} of R over X and Y.For example, ,=, and ,neq, are each other's complement, as are ,subseteq, and ,notsubseteq,, ,supseteq, and ,notsupseteq,, and ,in, and ,notin,, and, for total orders, also ,, and ,leq.,The complement of the converse relation R^textsf{T} is the converse of the complement: overline{R^mathsf{T}} = bar{R}^mathsf{T}.If X = Y, 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

If R is a binary homogeneous relation over a set X and S is a subset of X then R_{vert S} = { (x, y) mid xRy text{ and } x in S text{ and } y in S } is the {{em|{{visible anchor|restriction relation|Restriction relation|Restriction of a homogeneous relation}}}} 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 S} = { (x, y) mid xRy text{ and } x in S } is the {{em|{{visible anchor|left-restriction relation|Left-restriction relation}}}} of R to S over X and Y.{{clarify|reason=Introduce notational distinction between restriction and left restriction.|date=November 2022}}If R is a binary relation over sets X and Y and if S is a subset of Y then R^{vert S} = { (x, y) mid xRy text{ and } y in S } is the {{em|{{visible anchor|right-restriction relation|Right-restriction relation}}}} 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 {{em|{{visible anchor|contained in|Containment of relations}}}} a relation S over X and Y, written R subseteq S, if R is a subset of S, that is, for all x in X and y in Y, if xRy, then xSy. If R is contained in S and S is contained in R, then R and S are called {{em|equal}} written R = S. If R is contained in S but S is not contained in R, then R is said to be {{em|{{visible anchor|smaller|Smaller relation}}}} than S, written R subsetneq S. 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),NEWSGROUP, quantum mechanics over a commutative rig, John C. Baez, John C. Baez, 6 Nov 2001, sci.physics.research, 9s87n0$iv5@gap.cco.caltech.edu,weblink November 25, 2018, 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 {{math|1=X = Y}}) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. {{doi|10.1007/978-3-642-01492-5_1}}, pp. 7-10

Examples {| class"wikitable" style"float: right; margin-left:1em; text-align:center;"|+ 2nd example relation

! {{diagonal split header|{{math|B{{prime}}}}|{{math|A}}}}! scope="col" | ball! scope="col" | car! scope="col" | doll! scope="col" | cup! scope="row" | John ! scope="row" | Mary ! scope="row" | Venus

date=June 1970	url=https://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf	weblink >archive-date=2004-09-08	journal=Communications of the ACM	issue=6	doi=10.1145/362384.362685	access-date=2020-04-29, It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element {{mvar	related to an element {{mvar>y}}, if and only if the pair {{math	x, y)}} belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case {{math>1=n = 2}} of an Finitary relation	n}}-ary relation over sets {{math|X1, ..., Xn}}, which is a subset of the Cartesian product X_1 times cdots times X_n.An example of a binary relation is the "divides" relation over the set of prime numbers mathbb{P} and the set of integers mathbb{Z}, in which each prime {{mvar|p}} is related to each integer {{mvar|z}} that is a multiple of {{mvar|p}}, but not to an integer that is not a multiple of {{mvar|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: 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.WEB,weblink Relation definition – Math Insight, mathinsight.org, 2019-12-11, Binary relations are also heavily used in computer science.A binary relation over sets {{mvar|X}} and {{mvar|Y}} is an element of the power set of X times Y. Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of X times 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,Ernst Schröder (1895) Algebra und Logic der Relative, via Internet Archive Clarence Lewis,C. I. Lewis (1918) A Survey of Symbolic Logic, pages 269–279, via internet Archive and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called {{em|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.The terms {{em|correspondence}},Jacobson, Nathan (2009), Basic Algebra II (2nd ed.) § 2.1. 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 times Y without reference to {{mvar|X}} and {{mvar|Y}}, and reserve the term "correspondence" for a binary relation with reference to {{mvar|X}} and {{mvar|Y}}.{{citation needed|reason=Who?|date=June 2021}} Definition Given sets X and Y, the Cartesian product X times Y is defined as { (x, y) : x in X text{ and } y in Y }, and its elements are called ordered pairs.A {{em|binary relation}} R over sets X and Y is a subset of X times Y.{{harvnb|Enderton|1977|loc=Ch 3. pg. 40}} The set X is called the {{em|domain}} or {{em|set of departure}} of R, and the set Y the {{em|codomain}} or {{em|set of destination}} of R. In order to specify the choices of the sets X and Y, some authors define a {{em|binary relation}} or {{em|correspondence}} as an ordered triple {{math|(X, Y, G)}}, where G is a subset of X times Y called the {{em|graph}} of the binary relation. The statement (x, y) in R reads "x is R-related to y" and is denoted by xRy.{{#tag:ref|Authors who deal with binary relations only as a special case of n-ary relations for arbitrary n usually write Rxy as a special case of Rx1...xn (prefix notation).BOOK, 1431-4657, 3540058192, Hans Hermes, Introduction to Mathematical Logic, London, Springer, Hochschultext (Springer-Verlag), 1973, Sect.II.§1.1.4|group=note}} The {{em|domain of definition}} or {{em|active domain}} of R is the set of all x such that xRy for at least one y. The codomain of definition, {{em|active codomain}}, {{em|image}} or {{em|range}} of R is the set of all y such that xRy for at least one x. The {{em|field}} of R is the union of its domain of definition and its codomain of definition.BOOK , Suppes, Patrick, Patrick Suppes , 1972 , Axiomatic Set Theory , Dover , originally published by D. van Nostrand Company in 1960 , 0-486-61630-4 , registration ,weblink , 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 , 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 , When X = Y, a binary relation is called a {{em|homogeneous relation}} (or {{em|endorelation}}). To emphasize the fact that X and Y are allowed to be different, a binary relation is also called a heterogeneous relation.BOOK, Schmidt, Gunther, Ströhlein, Thomas, Relations and Graphs: Discrete Mathematics for Computer Scientists, {{google books, y, ZgarCAAAQBAJ, 277, |date=2012|publisher=Springer Science & Business Media|isbn=978-3-642-77968-8|author-link1=Gunther Schmidt |at=Definition 4.1.1.}}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,weblink BOOK, Michael Winter, Goguen Categories: A Categorical Approach to L-fuzzy Relations, 2007, Springer, 978-1-4020-6164-6, x-xi, In a binary relation, the order of the elements is important; if x neq 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 cup S = { (x, y) : xRy text{ or } xSy } is the {{em|union relation}} of R and S over X and Y.The identity element is the empty relation. For example, ,leq, is the union of
+ >		| −
		+ >| −
	+ >	| −

{| class="wikitable" style="float: right; margin-left:1em; text-align:center;"|+ 1st example relation! {{diagonal split header|{{math|B}}|{{math|A}}}}! scope="col" | ball! scope="col" | car! scope="col" | doll! scope="col" | cup! scope="row" | John + >| −! scope="row" | Mary + >| −! scope="row" | Ian | −! scope="row" | Venus + >| −{{olistA and B is the relation "is owned by", given by R = { (text{ball, John}), (text{doll, Mary}), (text{car, Venus}) }. That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing; see the 1st example. As a set, R does not involve Ian, and therefore R could have been viewed as a subset of A times { text{John, Mary, Venus} }, i.e. a relation over A and { text{John, Mary, Venus} }; see the 2nd example. While the 2nd example relation is surjective (see #Types of binary relations>below), the 1st is not.(File:Oceans and continents coarse.png|thumb|250px|right|Oceans and continents (islands omitted)){{pipe escape|{| class{{=}}"wikitable" style{{=}}"float: right; margin-left:1em; text-align:center;"|+Ocean borders continent!! scope{{=}}"col" | NA! scope{{=}}"col" | SA! scope{{=}}"col" | AF! scope{{=}}"col" | EU! scope{{=}}"col" | AS! scope{{=}}"col" | AU! scope{{=}}"col" | AA! scope{{=}}"row" | Indian|1 ! scope{{=}}"row" | Arctic|0 ! scope{{=}}"row" | Atlantic|1! scope{{=}}"row" | Pacific|1}}|2= Let A = {Indian, Arctic, Atlantic, Pacific}, the oceans of the globe, and B = { NA, SA, AF, EU, AS, AU, AA }, the continents. Let aRb represent that ocean a borders continent b. Then the logical matrix for this relation is: The connectivity of the planet Earth can be viewed through R RT and RT R, the former being a 4 times 4 relation on A, which is the universal relation (A times A or a logical matrix of all ones). This universal relation reflects the fact that every ocean is separated from the others by at most one continent. On the other hand, RT R is a relation on B times B which fails to be universal because at least two oceans must be traversed to voyage from Europe to Australia.graph theory: For relations on a set (homogeneous relations), a directed graph illustrates a relation and a graph (discrete mathematics)>graph a symmetric relation. For heterogeneous relations a hypergraph has edges possibly with more than two nodes, and can be illustrated by a bipartite graph.Just as the clique is integral to relations on a set, so bicliques are used to describe heterogeneous relations; indeed, they are the "concepts" that generate a lattice associated with a relation.(File:Add_velocity_ark_POV.svg|right|thumb|200px|The various t axes represent time for observers in motion, the corresponding x axes are their lines of simultaneity)Hyperbolic orthogonality: Time and space are different categories, and temporal properties are separate from spatial properties. The idea of {{em>simultaneous events}} is simple in absolute time and space since each time t determines a simultaneous hyperplane in that cosmology. Herman Minkowski changed that when he articulated the notion of {{em|relative simultaneity}}, which exists when spatial events are "normal" to a time characterized by a velocity. He used an indefinite inner product, and specified that a time vector is normal to a space vector when that product is zero. The indefinite inner product in a composition algebra is given by As a relation between some temporal events and some spatial events, hyperbolic orthogonality (as found in split-complex numbers) is a heterogeneous relation.geometric configuration can be considered a relation between its points and its lines. The relation is expressed as incidence relation>incidence. Finite and infinite projective and affine planes are included. Jakob Steiner pioneered the cataloguing of configurations with the Steiner systems text{S}(t, k, n) which have an n-element set S and a set of k-element subsets called blocks, such that a subset with t elements lies in just one block. These incidence structures have been generalized with block designs. The incidence matrix used in these geometrical contexts corresponds to the logical matrix used generally with binary relations. }}

Types of binary relations

File:The four types of binary relations.png|thumb|Examples of four types of binary relations over the (real number]]s: 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): for all x, y in X and all z in Y, if xRz and yRz then x = y. 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).
• UnivalentGunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, {{ISBN|978-0-521-76268-7}}, Chapt. 5 (also called right-unique, right-definite{{citation|title=Spatial Information Theory: 8th International Conference, COSIT 2007, Melbourne, Australia, September 19–23, 2007, Proceedings|series=Lecture Notes in Computer Science|publisher=Springer|volume=4736|year=2007|pages=285–302|contribution=Reasoning on Spatial Semantic Integrity Constraints|first=Stephan|last=Mäs|doi=10.1007/978-3-540-74788-8_18}} or functionalJOURNAL,weblink A. Sengupta, International Journal of Bifurcation and Chaos, 2003, Toward a Theory of Chaos, 10.1142/S021812740300851X, 13, 11, 3147–3233, ): for all x in X and all y, z in Y, if xRy and xRz then y = z. Such a binary relation is called a {{em|partial function}}. For such a relation, { X } is called {{em|a primary key}} of R. 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):Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:
• BOOK

,

• BOOK

,

• BOOK

, for all x in X there exists a y in Y such that xRy. In other words, the domain of definition of R is equal to X. This property, is different from the definition of {{em|connected}} (also called {{em|total}} by some authors){{citation needed|date=June 2020}} in Properties. Such a binary relation is called a {{em|multivalued function}}. 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, >

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 ,subseteq_A., Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation ,in_A, 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 {{math|(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.)BOOK, A formalization of set theory without variables, Tarski, Alfred, Alfred Tarski, Givant, Steven, 1987, 3, American Mathematical Society, 0-8218-1041-3,weblink With this definition one can for instance define a binary relation over every set and its power set.

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 times X.BOOK, M. E. Müller, Relational Knowledge Discovery, 2012, Cambridge University Press, 978-0-521-19021-3, 22, BOOK, Peter J. Pahl, Rudolf Damrath, Mathematical Foundations of Computational Engineering: A Handbook, 2001, Springer Science & Business Media, 978-3-540-67995-0, 496, 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 {{math|xRy}}).The set of all homogeneous relations mathcal{B}(X) over a set X is the power set 2^{X times X} 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 mathcal{B}(X), it forms a semigroup with involution.Some important properties that a homogeneous relation {{mvar|R}} over a set {{mvar|X}} may have are:

• {{em|Reflexive}}: for all x in X, {{math|xRx}}. For example, ,geq, is a reflexive relation but > is not.
• {{em|Irreflexive}}: for all x in X, not {{math|xRx}}. For example, ,>, is an irreflexive relation, but ,geq, is not.
• {{em|Symmetric}}: for all x, y in X, if {{math|xRy}} then {{math|yRx}}. For example, "is a blood relative of" is a symmetric relation.
• {{em|Antisymmetric}}: for all x, y in X, if {{math|xRy}} and {{math|yRx}} then x = y. For example, ,geq, is an antisymmetric relation.{{citation|first1=Douglas|last1=Smith|first2=Maurice|last2=Eggen|first3=Richard|last3=St. Andre|title=A Transition to Advanced Mathematics|edition=6th|publisher=Brooks/Cole|year=2006|isbn=0-534-39900-2|page=160}}
• {{em|Asymmetric}}: for all x, y in X, if {{math|xRy}} then not {{math|yRx}}. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.{{citation|first1=Yves|last1=Nievergelt|title=Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography|publisher=Springer-Verlag|year=2002|page=158}}. For example, > is an asymmetric relation, but ,geq, is not.
• {{em|Transitive}}: for all x, y, z in X, if {{math|xRy}} and {{math|yRz}} then {{math|xRz}}. A transitive relation is irreflexive if and only if it is asymmetric.BOOK, FlaÅ¡ka, V., Ježek, J., Kepka, T., Kortelainen, J., Transitive Closures of Binary Relations I, 2007, School of Mathematics â€“ Physics Charles University, Prague, 1,weblink dead,weblink" title="web.archive.org/web/20131102214049weblink">weblink 2013-11-02, Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric". For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
• {{em|Connected}}: for all x, y in X, if x neq y then {{math|xRy}} or {{math|yRx}}.
• {{em|Strongly connected}}: for all x, y in X, {{math|xRy}} or {{math|yRx}}.
• {{em|Dense}}: for all x, y in X, if xRy , then some z in X exists such that xRz and zRy.

A {{em|partial order}} is a relation that is reflexive, antisymmetric, and transitive. A {{em|strict partial order}} is a relation that is irreflexive, asymmetric, and transitive. A {{em|total order}} is a relation that is reflexive, antisymmetric, transitive and connected.Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, {{ISBN|0-12-597680-1}}, p. 4 A {{em|strict total order}} is a relation that is irreflexive, asymmetric, transitive and connected.An {{em|equivalence relation}} 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 {{slink||Operations}} also apply to homogeneous relations.Beyond that, a homogeneous relation over a set X may be subjected to closure operations like:

{{em|Reflexive closure}}: the smallest reflexive relation over X containing R,

{{em|Transitive closure}}: the smallest transitive relation over X containing R,

{{em|Equivalence closure}}: 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 A times B, where A and B are possibly distinct sets. The prefix hetero is from the Greek ἕτερος (heteros, "other, another, different").A heterogeneous relation has been called a rectangular relation, 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 {{em|heterogeneous}} or {{em|rectangular}}, i.e. as relations where the normal case is that they are relations between different sets."G. Schmidt, Claudia Haltensperger, and Michael Winter (1997) "Heterogeneous relation algebra", chapter 3 (pages 37 to 53) in Relational Methods in Computer Science, Advances in Computer Science, Springer books {{ISBN|3-211-82971-7}}

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 subseteq S, meaning that aRb implies aSb, sets the scene in a lattice of relations. But since P subseteq Q equiv (P cap bar{Q} = varnothing ) equiv (P cap Q = 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 times 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 {{em|objects}} of the category Rel are sets, and the relation-morphisms compose as required in a category.{{citation needed|reason=Who has suggested this, when, and where?|date=June 2021}}

Induced concept lattice

Binary relations have been described through their induced concept lattices:A concept C ⊂ R satisfies two properties:

• The logical matrix of C is the outer product of logical vectors C_{i j} = u_i v_j , quad u, v logical vectors.{{clarify|reason=Given R, how are the logical vectors obtained?|date=June 2021}}
• C is maximal, not contained in any other outer product. Thus C is described as a non-enlargeable rectangle.

For a given relation R subseteq X times 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".R. Berghammer & M. Winter (2013) "Decomposition of relations on concept lattices", Fundamenta Informaticae 126(1): 37–82 {{doi|10.3233/FI-2013-871}} The decomposition is 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.Ki Hang Kim (1982) Boolean Matrix Theory and Applications, page 37, Marcel Dekker {{ISBN|0-8247-1788-0}} Structural analysis of relations with concepts provides an approach for data mining.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 {{mr|id=2781235}}

Particular relations

• Proposition: If R is a serial relation and RT is its transpose, then I subseteq R^textsf{T} R where {{mvar|I}} is the m × m identity relation.
• Proposition: If R is a surjective relation, then I subseteq R R^textsf{T} where {{mvar|I}} is the n times n identity relation.

Difunctional

{{anchor|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 = F G^textsf{T} is a composition of relations using {{em|univalent}} relations F subseteq A times Z text{ and } G subseteq B times Z. Jacques Riguet named these relations difunctional since the composition F GT involves univalent relations, commonly called partial functions.In 1950 Rigeut showed that such relations satisfy the inclusion:JOURNAL, Riguet, Jacques, Jacques Riguet, Comptes rendus, January 1950,weblink fr, Quelques proprietes des relations difonctionelles, 230, 1999–2000, 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.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, More formally, a relation {{mvar|R}} on X times Y is difunctional if and only if it can be written as the union of Cartesian products A_i times B_i, where the A_i are a partition of a subset of {{mvar|X}} and the B_i likewise a partition of a subset of {{mvar|Y}}.JOURNAL, 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, 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 x_1 cap x_2 neq varnothing implies x_1 R = x_2 R.BOOK, Chris Brink, Wolfram Kahl, Gunther Schmidt, Relational Methods in Computer Science, 1997, Springer Science & Business Media, 978-3-211-82971-4, 200, In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management."Ali Jaoua, Nadin Belkhiter, Habib Ounalli, and Theodore Moukam (1997) "Databases", pages 197–210 in Relational Methods in Computer Science, edited by Chris Brink, Wolfram Kahl, and Gunther Schmidt, Springer Science & Business Media {{isbn|978-3-211-82971-4}} Furthermore, difunctional relations are fundamental in the study of bisimulations.BOOK, 10.1007/978-3-662-44124-4_7 title = Coalgebraic Methods in Computer Science pages = 118Lecture Notes in Computer Science> year = 2014 first1 = H. P. first2 = M., 978-3-662-44123-7, 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.J. Riguet (1951) "Les relations de Ferrers", Comptes Rendus 232: 1729,30The 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{R}^textsf{T} R subseteq R.If any one of the relations R, bar{R}, R^textsf{T} is of Ferrers type, then all of them are.BOOK, Schmidt, Gunther, Ströhlein, Thomas, Relations and Graphs: Discrete Mathematics for Computer Scientists, {{google books, y, ZgarCAAAQBAJ, 277, |date=2012|publisher=Springer Science & Business Media|isbn=978-3-642-77968-8|authorlink1=Gunther Schmidt |page=77}}

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:

1. text{for all } x in A, Y = { x } text{ implies } xgY.
2. Y subseteq Z text{ and } xgY text{ implies } xgZ.
3. text{for all } y in Y, ygZ text{ and } xgY text{ implies } xgZ.

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.JOURNAL,weblink Georg Aumann, Kontakt-Relationen, Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften München, 1970, II, 67–77, 1971, Anne K. Steiner (1970) Review:Kontakt-Relationen from Mathematical ReviewsIn terms of the calculus of relations, sufficient conditions for a contact relation includeC^textsf{T} bar{C} subseteq ni bar{C} equiv C overline{ni bar{C}} subseteq C, where ni is the converse of set membership ({{math|∈}}).{{rp|280}}

Preorder RR

Every relation R generates a preorder R backslash R which is the left residual.In this context, the symbol ,backslash, does not mean "set difference". In terms of converse and complements, R backslash R equiv overline{R^textsf{T} bar{R}}. Forming the diagonal of R^textsf{T} bar{R}, the corresponding row of R^{text{T}} and column of bar{R} will be of opposite logical values, so the diagonal is all zeros. Then To show transitivity, one requires that (Rbackslash R)(Rbackslash R) subseteq R backslash R. Recall that X = R backslash R is the largest relation such that R X subseteq R. Then 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:

Fringe of a relation

Given a relation R, a sub-relation called its {{em|fringe}} is defined asoperatorname{fringe}(R) = R cap overline{R bar{R}^textsf{T} R}.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 (R subseteq bar{I}) 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.Gunther Schmidt (2011) Relational Mathematics, pages 211−15, Cambridge University Press {{ISBN|978-0-521-76268-7}}

Mathematical heaps

Given two sets A and B, the set of binary relations between them mathcal{B}(A,B) can be equipped with a ternary operation [a, b, c] = a b^textsf{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.Viktor Wagner (1953) "The theory of generalised heaps and generalised groups", Matematicheskii Sbornik 32(74): 545 to 632 {{mr|id=0059267}}C.D. Hollings & M.V. Lawson (2017) Wagner's Theory of Generalised Heaps, Springer books {{ISBN|978-3-319-63620-7}} {{mr|id=3729305}} The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:{{Blockquote|text=There is a pleasant symmetry in Wagner's work between heaps, semiheaps, and generalised heaps on the one hand, and groups, semigroups, and generalised groups on the other. Essentially, the various types of semiheaps appear whenever we consider binary relations (and partial one-one mappings) between different sets A and B, while the various types of semigroups appear in the case where A = B.|author=Christopher HollingsMathematics across the Iron Curtain: a history of the algebraic theory of semigroups, page 265, History of Mathematics 41, American Mathematical Society {{ISBN>978-1-4704-1493-1}}}}

See also

{{div col}}

• Abstract rewriting system
• Additive relation, a many-valued homomorphism between modules
• Allegory (category theory)
• Category of relations, a category having sets as objects and binary relations as morphisms
• Confluence (term rewriting), discusses several unusual but fundamental properties of binary relations
• Correspondence (algebraic geometry), a binary relation defined by algebraic equations
• Hasse diagram, a graphic means to display an order relation
• Incidence structure, a heterogeneous relation between set of points and lines
• Logic of relatives, a theory of relations by Charles Sanders Peirce
• Order theory, investigates properties of order relations

{{colend}}

Notes

{{reflist|group=note}}

References

{{reflist}}

Bibliography

• BOOK, Schmidt, Gunther, Ströhlein, Thomas, Relations and Graphs: Discrete Mathematics for Computer Scientists, {{google books, y, ZgarCAAAQBAJ, 54, |date=2012|chapter=Chapter 3: Heterogeneous relations|publisher=Springer Science & Business Media|isbn=978-3-642-77968-8|authorlink1=Gunther Schmidt}}
• Ernst Schröder (1895) Algebra der Logik, Band III, via Internet Archive
• BOOK, Codd, Edgar Frank, Edgar F. Codd, 1990, The Relational Model for Database Management: Version 2,weblinkweblink 2022-10-09, live, Boston, Addison-Wesley, 978-0201141924,
• BOOK, Enderton, Herbert, Herbert Enderton, 1977, Elements of Set Theory, Boston, Academic Press, 978-0-12-238440-0,
• BOOK, Kilp, Mati, Knauer, Ulrich, Mikhalev, Alexander, 2000, Monoids, Acts and Categories: with Applications to Wreath Products and Graphs, Berlin, Walter de Gruyter, De Gruyter, 978-3-11-015248-7,
• JOURNAL, Peirce, Charles Sanders, Charles Sanders Peirce, 1873, Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic,weblink Memoirs of the American Academy of Arts and Sciences, 9, 2, 317–178, 10.2307/25058006, 25058006, 1873MAAAS...9..317P, 2027/hvd.32044019561034, 2020-05-05, free,
• BOOK, Schmidt, Gunther, Gunther Schmidt, 2010, Relational Mathematics,weblink Cambridge, Cambridge University Press, 978-0-521-76268-7,

External links

• {{commons category-inline|Binary relations}}
• {{springer|title=Binary relation|id=p/b016380}}

{{Order theory}}{{Set theory}}