Binary relation

This article covers advanced notions. For basic topics, seeRelation (mathematics).

Transitivebinary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Preorder(Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all$a,b$and$S\neq \varnothing:$ ${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$ ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$ ${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ $aRa$ ${\text{not }}aRa$ ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$
Yindicates that the column's property is always true the row's term (at the very left), while✗indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated byYin the "Symmetric" column and✗in the "Antisymmetric" column, respectively. All definitions tacitly require thehomogeneous relation$R$betransitive:for all$a,b,c,$if$aRb$and$bRc$then$aRc.$ A term's definition may require additional properties that are not listed in this table.

Inmathematics,abinary relationassociates elements of one set, called thedomain,with elements of another set, called thecodomain.A binary relation oversets$X$and$Y$is a set ofordered pairs$(x,y)$consisting of elements$x$from$X$and$y$from$Y$.It encodes the common concept of relation: an element$x$isrelatedto an element$y$,if and only ifthe 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 ofprime numbers$\mathbb {P}$and the set ofintegers$\mathbb {Z}$,in which each prime$p$is related to each integer$z$that is amultipleof$p$,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:

• the "is greater than","is equal to",and" divides "relations inarithmetic;
• the "is congruent to"relation ingeometry;
• the "is adjacent to" relation ingraph theory;
• the "isorthogonalto "relation inlinear algebra.

Afunctionmay be defined as a binary relation that meets additional constraints.Binary relations are also heavily used incomputer science.

A binary relation over sets$X$and$Y$is an element of thepower setof$X\times Y.$Since the latter set is ordered byinclusion($\subseteq$), each relation has a place in thelatticeof subsets of$X\times Y.$A binary relation is called ahomogeneous relationwhen$X=Y$.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, includingunion,intersection,andcomplementation,and satisfying the laws of analgebra of sets.Beyond that, operations like theconverseof a relation and thecomposition of relationsare available, satisfying the laws of acalculus of relations,for which there are textbooks byErnst Schröder,Clarence Lewis,andGunther Schmidt.A deeper analysis of relations involves decomposing them into subsets calledconcepts,and placing them in acomplete lattice.

In some systems ofaxiomatic set theory,relations are extended toclasses,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 asRussell's paradox.

A binary relation is the most studied special case$n=2$of an$n$-ary relationover sets$X_{1},\dots,X_{n}$,which is a subset of theCartesian product$X_{1}\times \cdots \times X_{n}.$

Definition

Given sets$X$and$Y$,theCartesian product$X\times Y$is defined as$\{(x,y)\mid x\in X{\text{ and }}y\in Y\},$and its elements are calledordered pairs.

Abinary relation$R$over sets$X$and$Y$is a subset of$X\times Y.$The set$X$is called thedomainorset of departureof$R$,and the set$Y$thecodomainorset of destinationof$R$.In order to specify the choices of the sets$X$and$Y$,some authors define abinary relationorcorrespondenceas an ordered triple$(X,Y,G)$,where$G$is a subset of$X\times Y$called thegraphof the binary relation. The statement$(x,y)\in R$reads "$x$is$R$-related to$y$"and is denoted by$xRy$.Thedomain of definitionoractive domainof$R$is the set of all$x$such that$xRy$for at least one$y$.Thecodomain of definition,active codomain,imageorrangeof$R$is the set of all$y$such that$xRy$for at least one$x$.Thefieldof$R$is the union of its domain of definition and its codomain of definition.

When$X=Y,$a binary relation is called ahomogeneous relation(orendorelation). To emphasize the fact that$X$and$Y$are allowed to be different, a binary relation is also called aheterogeneous relation.The prefixheterois from the Greek ἕτερος (heteros,"other, another, different" ).

A heterogeneous relation has been called arectangular relation,suggesting that it does not have the square-like symmetry of ahomogeneous relation on a setwhere$A=B.$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 asheterogeneousorrectangular,i.e. as relations where the normal case is that they are relations between different sets. "

The termscorrespondence,dyadic relationandtwo-place relationare 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$X$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$x\neq y$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$R\cup S=\{(x,y)\mid xRy{\text{ or }}xSy\}$is theunion relationof$R$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$R\cap S=\{(x,y)\mid xRy{\text{ and }}xSy\}$is theintersection relationof$R$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

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$S\circ R=\{(x,z)\mid {\text{ there exists }}y\in Y{\text{ such that }}xRy{\text{ and }}ySz\}$(also denoted by$R;S$) is thecomposition relationof$R$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 forcomposition 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

Main article:Converse relation

See also:Duality (order theory)

If$R$is a binary relation over sets$X$and$Y$then$R^{\textsf {T}}=\{(y,x)\mid xRy\}$is theconverse relation,also calledinverse relation,of$R$over$Y$and$X$.

For example,$=$is the converse of itself, as is$\neq$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 issymmetric.

Complement

Main article:Complementary relation

If$R$is a binary relation over sets$X$and$Y$then${\bar {R}}=\{(x,y)\mid \neg xRy\}$(also denoted by$\neg R$) is thecomplementary relationof$R$over$X$and$Y$.

For example,$=$and$\neq$are each other's complement, as are$\subseteq$and$\not \subseteq$,$\supseteq$and$\not \supseteq$,$\in$and$\not \in$,and fortotal ordersalso$<$and$\geq$,and$>$and$\leq$.

The complement of theconverse 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 astrict weak orderis a total preorder—and vice versa.

Restriction

Main article:Restriction (mathematics)

If$R$is a binaryhomogeneous relationover 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 therestriction relationof$R$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 theleft-restriction relationof$R$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$R^{\vert S}=\{(x,y)\mid xRy{\text{ and }}y\in S\}$is theright-restriction relationof$R$to$S$over$X$and$Y$.

If a relation isreflexive,irreflexive,symmetric,antisymmetric,asymmetric,transitive,total,trichotomous,apartial order,total order,strict weak order,total preorder(weak order), or anequivalence 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 ofcompleteness(not to be confused with being "total" ) do not carry over to restrictions. For example, over thereal numbersa property of the relation$\leq$is that everynon-emptysubset$S\subseteq \mathbb {R}$with anupper boundin$\mathbb {R}$has aleast upper bound(also called supremum) in$\mathbb {R}.$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 becontained ina 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 calledequalwritten$R=S$.If$R$is contained in$S$but$S$is not contained in$R$,then$R$is said to besmallerthan$S$,written$R\subsetneq S.$For example, on therational 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 bylogical matricesindexed by$X$and$Y$with entries in theBoolean semiring(addition corresponds to OR and multiplication to AND) wherematrix additioncorresponds to union of relations,matrix multiplicationcorresponds to composition of relations (of a relation over$X$and$Y$and a relation over$Y$and$Z$),theHadamard productcorresponds to intersection of relations, thezero matrixcorresponds to the empty relation, and thematrix of onescorresponds to the universal relation. Homogeneous relations (when$X=Y$) form amatrix semiring(indeed, amatrix semialgebraover the Boolean semiring) where theidentity matrixcorresponds to the identity relation.

Examples

2nd example relation

$A$ $B$	ball	car	doll	cup
John	+	−	−	−
Mary	−	−	+	−
Venus	−	+	−	−

1st example relation

$A$ $B$	ball	car	doll	cup
John	+	−	−	−
Mary	−	−	+	−
Ian	−	−	−	−
Venus	−	+	−	−

1. The following example shows that the choice of codomain is important. Suppose there are four objects$A=\{{\text{ball, car, doll, cup}}\}$and four people$B=\{{\text{John, Mary, Ian, Venus}}\}.$A possible relation on$A$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. But in that second example,$R$contains no information about the ownership by Ian.

   While the 2nd example relation is surjective (seebelow), the 1st is not.

   

   Oceans and continents (islands omitted)

   Ocean borders continent

   	NA	SA	AF	EU	AS	AU	AA
   Indian	0	0	1	0	1	1	1
   Arctic	1	0	0	1	1	0	0
   Atlantic	1	1	1	1	0	0	1
   Pacific	1	1	0	0	1	1	1

2. Let$A=\{{\text{Indian}},{\text{Arctic}},{\text{Atlantic}},{\text{Pacific}}\}$,theoceansof the globe, and$B=\{{\text{NA}},{\text{SA}},{\text{AF}},{\text{EU}},{\text{AS}},{\text{AU}},{\text{AA}}\}$,thecontinents.Let$aRb$represent that ocean$a$borders continent$b$.Then thelogical matrixfor this relation is:

   $R={\begin{pmatrix}0&0&1&0&1&1&1\\1&0&0&1&1&0&0\\1&1&1&1&0&0&1\\1&1&0&0&1&1&1\end{pmatrix}}.$

   The connectivity of the planet Earth can be viewed through$RR^{\mathsf {T}}$and$R^{\mathsf {T}}R$,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,$R^{\mathsf {T}}R$is a relation on$B\times B$whichfailsto be universal because at least two oceans must be traversed to voyage fromEuropetoAustralia.
3. Visualization of relations leans ongraph theory:For relations on a set (homogeneous relations), adirected graphillustrates a relation and agraphasymmetric relation.For heterogeneous relations ahypergraphhas edges possibly with more than two nodes, and can be illustrated by abipartite graph. Just as thecliqueis integral to relations on a set, sobicliquesare used to describe heterogeneous relations; indeed, they are the "concepts" that generate a lattice associated with a relation.

   

   The various$t$axes represent time for observers in motion, the corresponding$x$axes are their lines of simultaneity

4. Hyperbolic orthogonality:Time and space are different categories, and temporal properties are separate from spatial properties. The idea ofsimultaneous eventsis simple inabsolute time and spacesince each time$t$determines a simultaneoushyperplanein that cosmology.Hermann Minkowskichanged that when he articulated the notion ofrelative 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 acomposition algebrais given by

   $\langle x,z\rangle =x{\bar {z}}+{\bar {x}}z\;$where the overbar denotes conjugation.

   As a relation between some temporal events and some spatial events,hyperbolic orthogonality(as found insplit-complex numbers) is a heterogeneous relation.
5. Ageometric configurationcan be considered a relation between its points and its lines. The relation is expressed asincidence.Finite and infinite projective and affine planes are included.Jakob Steinerpioneered the cataloguing of configurations with theSteiner systems$\operatorname {S} (t,k,n)$which have an n-element set$\operatorname {S}$and a set of k-element subsets calledblocks,such that a subset with$t$elements lies in just one block. Theseincidence structureshave been generalized withblock designs.Theincidence matrixused in these geometrical contexts corresponds to the logical matrix used generally with binary relations.

   An incidence structure is a triple$\mathbf {D} =(V,\mathbf {B},I)$where$V$and$\mathbf {B}$are any two disjoint sets and$I$is a binary relation between$V$and$\mathbf {B}$,i.e.$I\subseteq V\times \mathbf {B}.$The elements of$V$will be calledpoints,those of$\mathbf {B}$blocks,and those of$I$flags.

Types of binary relations

Examples of four types of binary relations over thereal 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 calledleft-unique): for all$x,y\in X$and all$z\in Y,$if$xRz$and$yRz$then$x=y$.In other words, every element of the codomain hasat mostonepreimageelement. For such a relation,$Y$is calledaprimary keyof$R$.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$0$).
• Functional(also calledright-uniqueorunivalent): for all$x\in X$and all$y,z\in Y,$if$xRy$and$xRz$then$y=z$.In other words, every element of the domain hasat mostoneimageelement. Such a binary relation is called apartial functionorpartial mapping.For such a relation,$\{X\}$is calleda primary keyof$R$.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$1$).
• 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 calledleft-total): for all$x\in X$there exists a$y\in Y$such that$xRy$.In other words, every element of the domain hasat leastone image element. In other words, the domain of definition of$R$is equal to$X$.This property, is different from the definition ofconnected(also calledtotalby some authors)inProperties.Such a binary relation is called amultivalued 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,$>$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$1>y$.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$y=x$.
• Surjective(also calledright-total): for all$y\in Y$,there exists an$x\in X$such that$xRy$.In other words, every element of the codomain hasat leastone 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):

• Afunction(also calledmapping): a binary relation that is functional and total. In other words, every element of the domain hasexactlyone image element. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
• Aninjection: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.
• Asurjection: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.
• Abijection:a function that is injective and surjective. In other words, every element of the domain hasexactlyone image element and every element of the codomain hasexactlyone preimage element. For example, the green binary relation in the diagram is a bijection, but the red, blue and black ones are not.

If relations over proper classes are allowed:

• Set-like(also calledlocal): for all$x\in X$,theclassof all$y\in Y$such that$yRx$,i.e.$\{y\in Y,yRx\}$,is a set. For example, the relation$\in$is set-like, and every relation on two sets is set-like.The usual ordering < over the class ofordinal numbersis 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 ofaxiomatic 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 innaive set theory,seeRussell's paradox.

Another solution to this problem is to use a set theory with proper classes, such asNBGorMorse–Kelley set theory,and allow the domain and codomain (and so the graph) to beproper 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.)With this definition one can for instance define a binary relation over every set and its power set.

Homogeneous relation

Main article:Homogeneous relation

Ahomogeneous relationover a set$X$is a binary relation over$X$and itself, i.e. it is a subset of the Cartesian product$X\times X.$It is also simply called a (binary) relation over$X$.

A homogeneous relation$R$over a set$X$may be identified with adirected 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${\mathcal {B}}(X)$over a set$X$is thepower set$2^{X\times X}$which is aBoolean algebraaugmented with theinvolutionof mapping of a relation to itsconverse relation.Consideringcomposition of relationsas abinary operationon${\mathcal {B}}(X)$,it forms asemigroup with involution.

Some important properties that a homogeneous relation$R$over a set$X$may have are:

• Reflexive:for all$x\in X,$$xRx$.For example,$\geq$is a reflexive relation but > is not.
• Irreflexive:for all$x\in X,$not$xRx$.For example,$>$is an irreflexive relation, but$\geq$is not.
• Symmetric:for all$x,y\in X,$if$xRy$then$yRx$.For example, "is a blood relative of" is a symmetric relation.
• Antisymmetric:for all$x,y\in X,$if$xRy$and$yRx$then$x=y.$For example,$\geq$is an antisymmetric relation.
• Asymmetric:for all$x,y\in X,$if$xRy$then not$yRx$.A relation is asymmetric if and only if it is both antisymmetric and irreflexive.For example, > is an asymmetric relation, but$\geq$is not.
• Transitive:for all$x,y,z\in X,$if$xRy$and$yRz$then$xRz$.A transitive relation is irreflexive if and only if it is asymmetric.For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
• Connected:for all$x,y\in X,$if$x\neq y$then$xRy$or$yRx$.
• Strongly connected:for all$x,y\in X,$$xRy$or$yRx$.
• Dense:for all$x,y\in X,$if$xRy,$then some$z\in X$exists such that$xRz$and$zRy$.

Apartial orderis a relation that is reflexive, antisymmetric, and transitive. Astrict partial orderis a relation that is irreflexive, asymmetric, and transitive. Atotal orderis a relation that is reflexive, antisymmetric, transitive and connected.Astrict total orderis a relation that is irreflexive, asymmetric, transitive and connected. Anequivalence relationis a relation that is reflexive, symmetric, and transitive. For example, "$x$divides$y$"is a partial, but not a total order onnatural numbers$\mathbb {N},$"$x<y$"is a strict total order on$\mathbb {N},$and "$x$is parallel to$y$"is an equivalence relation on the set of all lines in theEuclidean plane.

All operations defined in section§ Operationsalso apply to homogeneous relations. Beyond that, a homogeneous relation over a set$X$may be subjected to closure operations like:

Reflexive closure

the smallest reflexive relation over$X$containing$R$,

Transitive closure

the smallest transitive relation over$X$containing$R$,

Equivalence closure

the smallestequivalence relationover$X$containing$R$.

Calculus of relations

Developments inalgebraic logichave facilitated usage of binary relations. Thecalculus of relationsincludes thealgebra of sets,extended bycomposition of relationsand the use ofconverse relations.The inclusion$R\subseteq S,$meaning that$aRb$implies$aSb$,sets the scene in alatticeof 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 toSchröder rules,provides a calculus to work in thepower setof$A\times B.$

In contrast to homogeneous relations, thecomposition of relationsoperation is only apartial 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 ofcategory theoryas in thecategory of sets,except that themorphismsof this category are relations. Theobjectsof the categoryRelare sets, and the relation-morphisms compose as required in acategory.

Induced concept lattice

Binary relations have been described through their inducedconcept lattices: Aconcept$C\subset R$satisfies two properties:

• Thelogical matrixof$C$is theouter productof logical vectors$C_{ij}=u_{i}v_{j},\quad u,v$logical vectors.
• $C$is maximal, not contained in any other outer product. Thus$C$is described as anon-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 apreorder.

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

$R=fEg^{\textsf {T}}$,where$f$and$g$arefunctions,calledmappingsor 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 ofequivalence relationon a set.

Relations may be ranked by theSchein rankwhich counts the number of concepts necessary to cover a relation.Structural analysis of relations with concepts provides an approach fordata mining.

Particular relations

• Proposition:If$R$is aserial relationand$R^{\mathsf {T}}$is its transpose, then$I\subseteq R^{\textsf {T}}R$where$I$is the$m\times m$identity relation.
• Proposition:If$R$is asurjective relation,then$I\subseteq RR^{\textsf {T}}$where$I$is the$n\times n$identity relation.

Difunctional

The idea of a difunctional relation is to partition objects by distinguishing attributes, as a generalization of the concept of anequivalence relation.One way this can be done is with an intervening set$Z=\{x,y,z,\ldots \}$ofindicators.The partitioning relation$R=FG^{\textsf {T}}$is acomposition of relationsusingfunctionalrelations$F\subseteq A\times Z{\text{ and }}G\subseteq B\times Z.$Jacques Riguetnamed these relationsdifunctionalsince the composition$FG^{\mathsf {T}}$involves functional relations, commonly calledpartial functions.

In 1950 Riguet showed that such relations satisfy the inclusion:

$$RR^{\textsf {T}}R\subseteq R$$

Inautomata theory,the termrectangular relationhas also been used to denote a difunctional relation. This terminology recalls the fact that, when represented as alogical matrix,the columns and rows of a difunctional relation can be arranged as ablock matrixwith rectangular blocks of ones on the (asymmetric) main diagonal.More formally, a relation$R$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$X$and the$B_{i}$likewise a partition of a subset of$Y$.

Using the notation$\{y\mid xRy\}=xR$,a difunctional relation can also be characterized as a relation$R$such that wherever$x_{1}R$and$x_{2}R$have a non-empty intersection, then these two sets coincide; formally$x_{1}\cap x_{2}\neq \varnothing$implies$x_{1}R=x_{2}R.$

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies indatabasemanagement. "Furthermore, difunctional relations are fundamental in the study ofbisimulations.

In the context of homogeneous relations, apartial equivalence relationis difunctional.

Ferrers type

Astrict orderon a set is a homogeneous relation arising inorder theory. In 1951Jacques Riguetadopted the ordering of aninteger partition,called aFerrers diagram,to extend ordering to binary relations in general.

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 $$R{\bar {R}}^{\textsf {T}}R\subseteq R.$$

If any one of the relations$R,{\bar {R}},R^{\textsf {T}}$is of Ferrers type, then all of them are.

Contact

Suppose$B$is thepower setof$A$,the set of allsubsetsof$A$.Then a relation$g$is acontact relationif 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. ${\text{for all }}y\in Y,ygZ{\text{ and }}xgY{\text{ implies }}xgZ.$

Theset membershiprelation,$\epsilon =$"is an element of", satisfies these properties so$\epsilon$is a contact relation. The notion of a general contact relation was introduced byGeorg Aumannin 1970.

In terms of the calculus of relations, sufficient conditions for a contact relation include $$C^{\textsf {T}}{\bar {C}}\subseteq \ni {\bar {C}}\equiv C{\overline {\ni {\bar {C}}}}\subseteq C,$$ where$\ni$is the converse of set membership ($\in$).

Preorder R\R

Every relation$R$generates apreorder$R\backslash R$which is theleft residual.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$R^{\textsf {T}}$and column of${\bar {R}}$will be of opposite logical values, so the diagonal is all zeros. Then

$R^{\textsf {T}}{\bar {R}}\subseteq {\bar {I}}\implies I\subseteq {\overline {R^{\textsf {T}}{\bar {R}}}}=R\backslash R$,so that$R\backslash R$is areflexive relation.

To showtransitivity,one requires that$(R\backslash R)(R\backslash R)\subseteq R\backslash R.$Recall that$X=R\backslash R$is the largest relation such that$RX\subseteq R.$Then

$R(R\backslash R)\subseteq R$

$R(R\backslash R)(R\backslash R)\subseteq R$(repeat)

$\equiv R^{\textsf {T}}{\bar {R}}\subseteq {\overline {(R\backslash R)(R\backslash R)}}$(Schröder's rule)

$\equiv (R\backslash R)(R\backslash R)\subseteq {\overline {R^{\textsf {T}}{\bar {R}}}}$(complementation)

$\equiv (R\backslash R)(R\backslash R)\subseteq R\backslash R.$(definition)

Theinclusionrelation Ω on thepower setof$U$can be obtained in this way from themembership relation$\in$on subsets of$U$:

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

Fringe of a relation

Given a relation$R$,itsfringeis the sub-relation defined as $$\operatorname {fringe} (R)=R\cap {\overline {R{\bar {R}}^{\textsf {T}}R}}.$$

When$R$is a partial identity relation, difunctional, or a block diagonal relation, then$\operatorname {fringe} (R)=R$.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 triangularlinear orderorstrict 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 adense,linear, strict order.

Mathematical heaps

Main article:Heap (mathematics)

Given two sets$A$and$B$,the set of binary relations between them${\mathcal {B}}(A,B)$can be equipped with aternary operation$[a,b,c]=ab^{\textsf {T}}c$where$b^{\mathsf {T}}$denotes theconverse relationof$b$.In 1953Viktor Wagnerused properties of this ternary operation to define semiheaps, heaps, and generalized heaps.The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:

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) betweendifferentsets$A$and$B$,while the various types of semigroups appear in the case where$A=B$.

— Christopher Hollings, "Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups"

See also

• 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

Bibliography

• Schmidt, Gunther (2010).Relational Mathematics.Berlin: Cambridge University Press.ISBN9780511778810.
• Schmidt, Gunther;Ströhlein, Thomas (2012)."Chapter 3: Heterogeneous relations".Relations and Graphs: Discrete Mathematics for Computer Scientists.Springer Science & Business Media.ISBN978-3-642-77968-8.
• Ernst Schröder(1895)Algebra der Logik, Band III,viaInternet Archive
• Codd, Edgar Frank(1990).The Relational Model for Database Management: Version 2(PDF).Boston:Addison-Wesley.ISBN978-0201141924.Archived(PDF)from the original on 2022-10-09.
• Enderton, Herbert(1977).Elements of Set Theory.Boston:Academic Press.ISBN978-0-12-238440-0.
• Kilp, Mati; Knauer, Ulrich; Mikhalev, Alexander (2000).Monoids, Acts and Categories: with Applications to Wreath Products and Graphs.Berlin:De Gruyter.ISBN978-3-11-015248-7.
• Van Gasteren, Antonetta (1990).On the Shape of Mathematical Arguments.Berlin: Springer.ISBN9783540528494.
• Peirce, Charles Sanders(1873)."Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic".Memoirs of the American Academy of Arts and Sciences.9(2): 317–178.Bibcode:1873MAAAS...9..317P.doi:10.2307/25058006.hdl:2027/hvd.32044019561034.JSTOR25058006.Retrieved2020-05-05.
• Schmidt, Gunther(2010).Relational Mathematics.Cambridge:Cambridge University Press.ISBN978-0-521-76268-7.

External links

• Media related toBinary relationsat Wikimedia Commons
• "Binary relation",Encyclopedia of Mathematics,EMS Press,2001 [1994]