View source for Binary relation

{{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 (mathematics)|set]] called the ''domain'' with elements of another set called the ''codomain''.<ref>{{Cite web|last=Meyer|first=Albert|date=17 November 2021|title=MIT 6.042J Math for Computer Science, Lecture 3T, Slide 2|url=https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2015/lecture-slides/MIT6_042JS16_Relations.pdf|url-status=live|archive-url=https://web.archive.org/web/20211117161447/https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2015/lecture-slides/MIT6_042JS16_Relations.pdf |archive-date=2021-11-17 }}</ref> Precisely, a binary relation over sets <math>X</math> and <math>Y</math> is a set of [[ordered pair]]s <math>(x, y)</math> where <math>x</math> is in <math>X</math> and <math>y</math> is in <math>Y</math>.<ref name="Codd1970">{{cite journal |last1=Codd |first1=Edgar Frank 
|authorlink=Edgar F. Codd|date=June 1970 |title=A Relational Model of Data for Large Shared Data Banks |url=https://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf |archive-url=https://web.archive.org/web/20040908011134/http://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf |archive-date=2004-09-08 |url-status=live |journal=Communications of the ACM |volume=13 |issue=6 |pages=377–387 |doi=10.1145/362384.362685 |s2cid=207549016 |access-date=2020-04-29}}</ref> It encodes the common concept of relation: an element <math>x</math> is ''related'' to an element <math>y</math>, [[if and only if]] the pair <math>(x, y)</math> belongs to the set of ordered pairs that defines the binary relation.

An example of a binary relation is the "[[divides]]" relation over the set of [[prime number]]s <math>\mathbb{P}</math> and the set of [[integer]]s <math>\mathbb{Z}</math>, in which each prime <math>p</math> is related to each integer <math>z</math> that is a [[Divisibility|multiple]] of <math>p</math>, but not to an integer that is not a [[Multiple (mathematics)|multiple]] of <math>p</math>. In this relation, for instance, the prime number <math>2</math> is related to numbers such as <math>-4</math>, <math>0</math>, <math>6</math>, <math>10</math>, but not to <math>1</math> or <math>9</math>, just as the prime number <math>3</math> is related to <math>0</math>, <math>6</math>, and <math>9</math>, but not to <math>4</math> or <math>13</math>.

Binary relations, and especially [[homogeneous relation]]s, are used in many branches of mathematics to model a wide variety of concepts. These include, among others:
* the "[[Inequality (mathematics)|is greater than]]", "[[Equality (mathematics)|is equal to]]", and "divides" relations in [[arithmetic]];
* the "[[Congruence (geometry)|is congruent to]]" relation in [[geometry]];
* the "is adjacent to" relation in [[graph theory]];
* the "is [[orthogonal]] to" relation in [[linear algebra]].

A [[Function (mathematics)|function]] may be defined as a binary relation that meets additional constraints.<ref>{{Cite web|url=https://mathinsight.org/definition/relation|title=Relation definition – Math Insight|website=mathinsight.org|access-date=2019-12-11}}</ref> Binary relations are also heavily used in [[computer science]].

A binary relation over sets <math>X</math> and <math>Y</math> is an element of the [[power set]] of <math>X \times Y.</math> Since the latter set is ordered by [[Inclusion (set theory)|inclusion]] (<math>\subseteq</math>), each relation has a place in the [[Lattice (order)|lattice]] of subsets of <math>X \times Y.</math> A binary relation is called a [[#Homogeneous relation|''homogeneous relation'']] when <math>X = Y</math>. A binary relation is also called a ''heterogeneous relation'' when it is not necessary that <math>X = Y</math>.

Since relations are sets, they can be manipulated using set operations, including [[Union (set theory)|union]], [[Intersection (set theory)|intersection]], and [[Complement (set theory)|complementation]], and satisfying the laws of an [[algebra of sets]]. Beyond that, operations like the [[converse relation|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 (mathematician)|Ernst Schröder]],<ref name="Schroder.1895">[[Ernst Schröder (mathematician)|Ernst Schröder]] (1895) [https://archive.org/details/vorlesungenberd03mlgoog Algebra und Logic der Relative], via [[Internet Archive]]</ref> [[Clarence Lewis]],<ref name="Lewis.1918">[[C. I. Lewis]] (1918) [https://archive.org/details/asurveyofsymboli00lewiuoft A Survey of Symbolic Logic], pages 269–279, via internet Archive</ref> and [[Gunther Schmidt]].<ref name=gs>[[Gunther Schmidt]], 2010. ''Relational Mathematics''. Cambridge University Press, {{ISBN|978-0-521-76268-7}}, Chapt. 5</ref> A deeper analysis of relations involves decomposing them into subsets called ''concepts'', and placing them in a [[complete lattice]].

In some systems of [[axiomatic set theory]], relations are extended to [[Class (mathematics)|classes]], which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as [[Russell's paradox]].

A binary relation is the most studied special case <math>n = 2</math> of an [[Finitary relation|<math>n</math>-ary relation]] over sets <math>X_1, \dots, X_n</math>, which is a subset of the [[Cartesian product]] <math>X_1 \times \cdots \times X_n.</math><ref name="Codd1970"/>

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

A {{em|binary relation}} <math>R</math> over sets <math>X</math> and <math>Y</math> is a subset of <math>X \times Y.</math><ref name="Codd1970" /><ref>{{harvnb|Enderton|1977|loc=Ch 3. pg. 40}}</ref> The set <math>X</math> is called the {{em|domain}}<ref name="Codd1970" /> or {{em|set of departure}} of <math>R</math>, and the set <math>Y</math> the {{em|codomain}} or {{em|set of destination}} of <math>R</math>. In order to specify the choices of the sets <math>X</math> and <math>Y</math>, some authors define a {{em|binary relation}} or {{em|correspondence}} as an ordered triple <math>(X, Y, G)</math>, where <math>G</math> is a subset of <math>X \times Y</math> called the {{em|graph}} of the binary relation. The statement <math>(x, y) \in R</math> reads "<math>x</math> is <math>R</math>-related to <math>y</math>" and is denoted by <math>xRy</math>.<ref name="Schroder.1895"/><ref name="Lewis.1918"/><ref name=gs/>{{#tag:ref|Authors who deal with binary relations only as a special case of <math>n</math>-ary relations for arbitrary <math>n</math> usually write <math>Rxy</math> as a special case of <math>Rx_1\dots x_n</math> ([[Polish notation|prefix notation]]).<ref>{{cite book | issn=1431-4657 | isbn=3540058192 | author=Hans Hermes | title=Introduction to Mathematical Logic | location=London | publisher=Springer | series=Hochschultext (Springer-Verlag) | year=1973 }} Sect.II.§1.1.4</ref>|group=note}} The {{em|domain of definition}} or {{em|active domain}}<ref name="Codd1970" /> of <math>R</math> is the set of all <math>x</math> such that <math>xRy</math> for at least one <math>y</math>. The ''codomain of definition'', {{em|active codomain}},<ref name="Codd1970" /> {{em|image}} or {{em|range}} of <math>R</math> is the set of all <math>y</math> such that <math>xRy</math> for at least one <math>x</math>. The {{em|field}} of <math>R</math> is the union of its domain of definition and its codomain of definition.<ref name="suppes">
{{cite book
 |last=Suppes |first=Patrick |author-link=Patrick Suppes
 |year=1972
 |title=Axiomatic Set Theory
 |publisher=Dover
 |orig-year=originally published by D. van Nostrand Company in 1960
 |isbn=0-486-61630-4
 |url-access=registration
 |url=https://archive.org/details/axiomaticsettheo00supp_0
}}
</ref><ref name="smullyan">
{{cite book
 |last1=Smullyan |first1=Raymond M. |author-link=Raymond Smullyan
 |last2=Fitting |first2=Melvin
 |year=2010
 |title=Set Theory and the Continuum Problem
 |publisher=Dover
 |orig-year=revised and corrected republication of the work originally published in 1996 by Oxford University Press, New York
 |isbn=978-0-486-47484-7
}}
</ref><ref name="levy">
{{cite book
 |last=Levy |first=Azriel |author-link=Azriel Levy
 |year=2002
 |title=Basic Set Theory
 |publisher=Dover
 |orig-year=republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979
 |isbn=0-486-42079-5
}}
</ref>

When <math>X = Y,</math> a binary relation is called a {{em|[[homogeneous relation]]}} (or {{em|endorelation}}). To emphasize the fact that <math>X</math> and <math>Y</math> are allowed to be different, a binary relation is also called a '''heterogeneous relation'''.<ref name="Schmidt">{{cite book|last1=Schmidt|first1=Gunther|last2=Ströhlein|first2=Thomas|title=Relations and Graphs: Discrete Mathematics for Computer Scientists|url={{google books |plainurl=y |id=ZgarCAAAQBAJ|paged=277}}|date=2012|publisher=Springer Science & Business Media|isbn=978-3-642-77968-8|author-link1=Gunther Schmidt |at=Definition 4.1.1.}}</ref><ref name="FloudasPardalos2008">{{cite book|author1=Christodoulos A. Floudas|author-link1=Christodoulos Floudas|author2=Panos M. Pardalos|title=Encyclopedia of Optimization|year=2008|publisher=Springer Science & Business Media|isbn=978-0-387-74758-3|pages=299–300|edition=2nd|url=https://books.google.com/books?id=1a6lSRbQ4YsC&q=relation}}</ref><ref name="Winter2007">{{cite book|author=Michael Winter|title=Goguen Categories: A Categorical Approach to L-fuzzy Relations|year=2007|publisher=Springer|isbn=978-1-4020-6164-6|pages=x-xi}}</ref> The prefix ''hetero'' is from the Greek ἕτερος (''heteros'', "other, another, different").

A heterogeneous relation has been called a '''rectangular relation''',<ref name="Winter2007"/> suggesting that it does not have the square-like symmetry of a [[#Homogeneous relation|homogeneous relation on a set]] where <math>A = B.</math> Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "...&nbsp;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."<ref>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}}</ref>

The terms ''correspondence'',<ref>Jacobson, Nathan (2009), [https://books.google.com/books?id=hn75exNZZ-EC&q=correspondence Basic Algebra II (2nd ed.)] §&nbsp;2.1.</ref> '''dyadic relation'''<!---[[Dyadic relation]]---> and '''two-place relation'''<!---[[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 <math>X</math> and <math>Y</math>, and reserve the term "correspondence" for a binary relation with reference to <math>X</math> and <math>Y</math>.{{citation needed|reason=Who?|date=June 2021}}

In a binary relation, the order of the elements is important; if <math>x \neq y</math> then <math>yRx</math> can be true or false independently of <math>xRy</math>. For example, <math>3</math> divides <math>9</math>, but <math>9</math> does not divide <math>3</math>.

== Operations ==
=== Union ===
<!---This definition should appear before the closure defs, which refer to it:--->
If <math>R</math> and <math>S</math> are binary relations over sets <math>X</math> and <math>Y</math> then <math>R \cup S = \{ (x, y) \mid xRy \text{ or } xSy \}</math> is the {{em|union relation}} of <math>R</math> and <math>S</math> over <math>X</math> and <math>Y</math>.

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 <math>R</math> and <math>S</math> are binary relations over sets <math>X</math> and <math>Y</math> then <math>R \cap S = \{ (x, y) \mid xRy \text{ and } xSy \}</math> is the {{em|intersection relation}} of <math>R</math> and <math>S</math> over <math>X</math> and <math>Y</math>.

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|Composition of relations}}
If <math>R</math> is a binary relation over sets <math>X</math> and <math>Y</math>, and <math>S</math> is a binary relation over sets <math>Y</math> and <math>Z</math> then <math>S \circ R = \{ (x, z) \mid \text{ there exists } y \in Y \text{ such that } xRy \text{ and } ySz \}</math> (also denoted by <math>R; S</math>) is the {{em|composition relation}} of <math>R</math> and <math>S</math> over <math>X</math> and <math>Z</math>.

The identity element is the identity relation. The order of <math>R</math> and <math>S</math> 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 <math>x</math> is the parent of <math>y</math> and <math>y</math> is the mother of <math>z</math>, then <math>x</math> is the maternal grandparent of <math>z</math>.

=== Converse ===
{{main|Converse relation}}
{{see also|Duality (order theory)}}
If <math>R</math> is a binary relation over sets <math>X</math> and <math>Y</math> then <math>R^\textsf{T} = \{ (y, x) \mid xRy \}</math> is the {{em|converse relation}},<ref>[[Garrett Birkhoff]] & Thomas Bartee (1970) ''Modern Applied Algebra'', page 35, McGraw-Hill</ref> also called {{em|inverse relation}},<ref>[[Mary P. Dolciani]] (1962) ''Modern Algebra: Structure and Method'', Book 2, page 339, Houghton Mifflin</ref> of <math>R</math> over <math>Y</math> and <math>X</math>.

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 relation|symmetric]].

=== Complement ===
{{main|Complementary relation}}
If <math>R</math> is a binary relation over sets <math>X</math> and <math>Y</math> then <math>\bar{R} = \{ (x, y) \mid \neg xRy \}</math> (also denoted by <math>\neg R</math>) is the {{em|complementary relation}} of <math>R</math> over <math>X</math> and <math>Y</math>.

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>, <math>\in</math> and <math>\not \in</math>, and for [[total order]]s 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 ===
{{main|Restriction (mathematics)}}
If <math>R</math> is a binary [[homogeneous relation]] over a set <math>X</math> and <math>S</math> is a subset of <math>X</math> then <math>R_{\vert S} = \{ (x, y) \mid xRy \text{ and } x \in S \text{ and } y \in S \}</math> is the {{em|{{visible anchor|restriction relation|Restriction relation|Restriction of a homogeneous relation}}}} of <math>R</math> to <math>S</math> over <math>X</math>.

If <math>R</math> is a binary relation over sets <math>X</math> and <math>Y</math> and if <math>S</math> is a subset of <math>X</math> then <math>R_{\vert S} = \{ (x, y) \mid xRy \text{ and } x \in S \}</math> is the {{em|{{visible anchor|left-restriction relation|Left-restriction relation}}}} of <math>R</math> to <math>S</math> over <math>X</math> and <math>Y</math>.{{clarify|reason=Introduce notational distinction between restriction and left restriction.|date=November 2022}}

If a relation is [[Reflexive relation|reflexive]], irreflexive, [[Symmetric relation|symmetric]], [[Antisymmetric relation|antisymmetric]], [[Asymmetric relation|asymmetric]], [[Transitive relation|transitive]], [[Serial relation|total]], [[Trichotomy (mathematics)|trichotomous]], a [[partial order]], [[total order]], [[strict weak order]], [[Strict weak order#Total preorders|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 "<math>x</math> is parent of <math>y</math>" to females yields the relation "<math>x</math> is mother of the woman <math>y</math>"; 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 (order theory)|completeness]] (not to be confused with being "total") do not carry over to restrictions. For example, over the [[real number]]s a property of the relation <math>\leq</math> is that every [[Empty set|non-empty]] subset <math>S \subseteq \R</math> with an [[upper bound]] in <math>\R</math> has a [[Supremum|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.

<!---This definition is needed by the closure defs, too, but maybe should better given in an earlier section(?):--->
A binary relation <math>R</math> over sets <math>X</math> and <math>Y</math> is said to be {{em|{{visible anchor|contained in|Containment of relations}}}} a relation <math>S</math> over <math>X</math> and <math>Y</math>, written <math>R \subseteq S,</math> if <math>R</math> is a subset of <math>S</math>, that is, for all <math>x \in X</math> and <math>y \in Y,</math> if <math>xRy</math>, then <math>xSy</math>. If <math>R</math> is contained in <math>S</math> and <math>S</math> is contained in <math>R</math>, then <math>R</math> and <math>S</math> are called {{em|equal}} written <math>R = S</math>. If <math>R</math> is contained in <math>S</math> but <math>S</math> is not contained in <math>R</math>, then <math>R</math> is said to be {{em|{{visible anchor|smaller|Smaller relation}}}} than <math>S</math>, written <math>R \subsetneq S.</math> For example, on the [[rational number]]s, the relation <math>></math> is smaller than <math>\geq</math>, and equal to the composition <math>> \circ ></math>.

=== Matrix representation ===
Binary relations over sets <math>X</math> and <math>Y</math> can be represented algebraically by [[Logical matrix|logical matrices]] indexed by <math>X</math> and <math>Y</math> 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 <math>X</math> and <math>Y</math> and a relation over <math>Y</math> and <math>Z</math>),<ref>{{cite newsgroup |title=quantum mechanics over a commutative rig |author=John C. Baez |author-link=John C. Baez |date=6 Nov 2001 |newsgroup=sci.physics.research |message-id=9s87n0$iv5@gap.cco.caltech.edu |url=https://groups.google.com/d/msg/sci.physics.research/VJNPMCfreao/TMKt9tFYNwEJ |access-date=November 25, 2018}}</ref> the [[Hadamard product (matrices)|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>X = Y</math>) form a [[matrix semiring]] (indeed, a [[matrix semialgebra]] over the Boolean semiring) where the [[identity matrix]] corresponds to the identity relation.<ref name="droste">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</ref>

== Examples ==
{| class="wikitable" style="float: right; margin-left:1em; text-align:center;"
|+ 2nd example relation
! {{diagonal split header|<math>B</math>|<math>A</math>}}
! scope="col" | ball
! scope="col" | car
! scope="col" | doll
! scope="col" | cup
|-
! scope="row" | John 
| '''+''' || − || − || −
|-
! scope="row" | Mary 
| − || − || '''+''' || −
|-
! scope="row" | Venus 
| − || '''+''' || − || −
|}
{| class="wikitable" style="float: right; margin-left:1em; text-align:center;"
|+ 1st example relation
! {{diagonal split header|<math>B</math>|<math>A</math>}}
! scope="col" | ball
! scope="col" | car
! scope="col" | doll
! scope="col" | cup
|-
! scope="row" | John 
| '''+''' || − || − || −
|-
! scope="row" | Mary 
| − || − || '''+''' || −
|-
! scope="row" | Ian 
| − || − || − || −
|-
! scope="row" | Venus 
| − || '''+''' || − || −
|}
{{olist
|1= The following example shows that the choice of codomain is important. Suppose there are four objects <math>A = \{ \text{ball, car, doll, cup} \}</math> and four people <math>B = \{ \text{John, Mary, Ian, Venus} \}.</math> A possible relation on <math>A</math> and <math>B</math> is the relation "is owned by", given by <math>R = \{ (\text{ball, John}), (\text{doll, Mary}), (\text{car, Venus}) \}.</math> 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, <math>R</math> does not involve Ian, and therefore <math>R</math> could have been viewed as a subset of <math>A \times \{ \text{John, Mary, Venus} \},</math> i.e. a relation over <math>A</math> and <math>\{ \text{John, Mary, Venus} \};</math> see the 2nd example. But in that second example, <math>R</math> contains no information about the ownership by Ian.

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
|0||0||1||0||1||1||1
|-  
! scope{{=}}"row" | Arctic
|1||0||0||1||1||0||0 
|-
! scope{{=}}"row" | Atlantic
|1||1||1||1||0||0||1
|-
! scope{{=}}"row" | Pacific
|1||1||0||0||1||1||1
|}
}}

|2= Let <math>A = \{\text{Indian}, \text{Arctic}, \text{Atlantic}, \text{Pacific}\}</math>, the [[ocean]]s of the globe, and <math>B = \{\text{NA}, \text{SA}, \text{AF}, \text{EU}, \text{AS}, \text{AU}, \text{AA}\}</math>, the [[continent]]s. Let <math>aRb</math> represent that ocean <math>a</math> borders continent <math>b</math>. Then the [[logical matrix]] for this relation is:
:<math>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} .</math>
The connectivity of the planet Earth can be viewed through <math>R R^\mathsf{T}</math> and <math>R^\mathsf{T} R</math>, the former being a <math>4 \times 4</math> relation on <math>A</math>, which is the universal relation (<math>A \times A</math> 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, <math>R^\mathsf{T} R</math> is a relation on <math>B \times B</math> which ''fails'' to be universal because at least two oceans must be traversed to voyage from [[Europe]] to [[Australia]].

|3= Visualization of relations leans on [[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 (graph theory)|clique]] is integral to relations on a set, so [[biclique]]s 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 <math>t</math> axes represent time for observers in motion, the corresponding <math>x</math> 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 of {{em|simultaneous events}} is simple in [[absolute time and space]] since each time <math>t</math> determines a simultaneous [[hyperplane]] in that cosmology. [[Hermann 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
:<math>\langle x, z\rangle = x \bar{z} + \bar{x}z\;</math> where the overbar denotes conjugation.
As a relation between some temporal events and some spatial events, [[hyperbolic orthogonality]] (as found in [[split-complex number]]s) is a heterogeneous relation.<ref>{{wikibooks inline|Calculus/Hyperbolic angle#Split-complex theory|Relative simultaneity}}</ref>

|5= A [[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 system]]s <math>\operatorname S(t, k, n)</math> which have an n-element set <math>\operatorname S</math> and a set of k-element subsets called '''blocks''', such that a subset with <math>t</math> elements lies in just one block. These [[incidence structure]]s have been generalized with [[block design]]s. The [[incidence matrix]] used in these geometrical contexts corresponds to the logical matrix used generally with binary relations.
: An incidence structure is a triple <math>\mathbf D = (V, \mathbf B, I)</math> where <math>V</math> and <math>\mathbf B</math> are any two disjoint sets and <math>I</math> is a binary relation between <math>V</math> and <math>\mathbf B</math>, i.e. <math>I \subseteq V \times \mathbf B.</math> The elements of <math>V</math> will be called {{em|points}}, those of <math>\mathbf B</math> {{em|blocks}}, and those of <math>I</math> {{em|flags}}.<ref>{{cite book|first1=Thomas|last1=Beth|first2=Dieter|last2=Jungnickel|authorlink2=Dieter Jungnickel|first3=Hanfried|last3=Lenz|authorlink3=Hanfried Lenz|title=Design Theory|publisher=[[Cambridge University Press]]|page=15|year=1986}}. 2nd ed. (1999) {{ISBN|978-0-521-44432-3}}</ref>
}}

== Types of binary relations ==
<!-- [[functional relation]] redirects to this section -->
[[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 <math>R</math> over sets <math>X</math> and <math>Y</math> are listed below.

Uniqueness properties:
* '''Injective'''<!---[[Injective relation]]---><ref name="vangasteren1990">Van Gasteren 1990, p. 45.</ref> (also called '''left-unique'''<!---[[Left-unique relation]]---><ref name="kilp2000">Kilp, Knauer, Mikhalev 2000, p. 3.</ref>): for all <math>x, y \in X</math> and all <math>z \in Y,</math> if <math>xRz</math> and <math>yRz</math> then <math>x = y</math>. In other words, every element of the codomain has ''at most'' one [[Image (mathematics)|preimage]] element. For such a relation, <math>Y</math> is called ''a [[primary key]]'' of <math>R</math>.<ref name="Codd1970" /> For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both <math>-1</math> and <math>1</math> to <math>1</math>), nor the black one (as it relates both <math>-1</math> and <math>1</math> to <math>0</math>).
* '''[[Functional relationship|Functional]]'''<!---[[Functional relation]]---><ref name="vangasteren1990" /><ref>{{Cite web|title=Functional relation - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Functional_relation|access-date=2024-06-13|website=encyclopediaofmath.org}}</ref><ref>{{Cite web|title=functional relation in nLab|url=https://ncatlab.org/nlab/show/functional+relation|access-date=2024-06-13|website=ncatlab.org}}</ref> (also called '''right-unique'''<!---[[Right-unique relation]]---><ref name="kilp2000" /> or '''univalent'''<ref>Schmidt 2010, p. 49.</ref>): for all <math>x \in X</math> and all <math>y, z \in Y,</math> if <math>xRy</math> and <math>xRz</math> then <math>y = z</math>. In other words, every element of the domain has ''at most'' one [[Image (mathematics)|image]] element. Such a binary relation is called a {{em|[[partial function]]}} or {{em|partial mapping}}.<ref>Kilp, Knauer, Mikhalev 2000, p. 4.</ref> For such a relation, <math>\{ X \}</math> is called {{em|a primary key}} of <math>R</math>.<ref name="Codd1970" /> For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates <math>1</math> to both <math>1</math> and <math>-1</math>), nor the black one (as it relates <math>0</math> to both <math>-1</math> and <math>1</math>).
* '''One-to-one'''<!---[[One-to-one relation]]--->: 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'''<!---[[One-to-many relation]]--->: 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'''<!---[[Many-to-one relation]]--->: 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'''<!---[[Many-to-many relation]]--->: 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 <math>X</math> and codomain <math>Y</math> are specified):
* '''[[Total relation|Total]]'''<!---[[Total relation]]---><ref name="vangasteren1990" /> (also called '''left-total'''<!---[[Left-total relation]]---><ref name="kilp2000" />): for all <math>x \in X</math> there exists a <math>y \in Y</math> such that <math>xRy</math>. In other words, every element of the domain has ''at least'' one image element. In other words, the domain of definition of <math>R</math> is equal to <math>X</math>. This property, is different from the definition of {{em|[[Connected relation|connected]]}} (also called {{em|total}} by some authors){{citation needed|date=June 2020}} in [[Homogeneous relation#Properties|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 <math>-1</math> to any real number), nor the black one (as it does not relate <math>2</math> to any real number). As another example, <math>></math> is a total relation over the [[Integer|integers]]. But it is not a total relation over the positive integers, because there is no <math>y</math> in the positive integers such that <math>1 > y</math>.<ref>{{cite journal|last = Yao|first = Y.Y.|author2=Wong, S.K.M.|title = Generalization of rough sets using relationships between attribute values|journal = Proceedings of the 2nd Annual Joint Conference on Information Sciences|year = 1995|pages = 30–33|url = http://www2.cs.uregina.ca/~yyao/PAPERS/relation.pdf}}.</ref> However, <math><</math> is a total relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is total: for a given <math>x</math>, choose <math>y = x</math>.
* '''Surjective'''<!---[[Surjective relation]]---><ref name="vangasteren1990" /> (also called '''right-total'''<!---[[Right-total relation]]---><ref name="kilp2000" />): for all <math>y \in Y</math>, there exists an <math>x \in X</math> such that <math>xRy</math>. In other words, every element of the codomain has ''at least'' one preimage element. In other words, the codomain of definition of <math>R</math> is equal to <math>Y</math>. 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 <math>-1</math>), nor the black one (as it does not relate any real number to <math>2</math>).

Uniqueness and totality properties (only definable if the domain <math>X</math> and codomain <math>Y</math> are specified):
* A '''[[Function (mathematics)|function]]''' (also called '''mapping'''<ref name="kilp2000" />): a binary relation that is functional and total. In other words, every element of the domain has ''exactly'' one image element. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
* An '''[[Injective function|injection]]''': a function that is injective. For example, the green relation in the diagram is an injection, but the red one is not; the black and the blue relation is not even a function.
* A '''[[Surjective function|surjection]]''': a function that is surjective. For example, the green relation in the diagram is a surjection, but the red one is not.
* A '''[[Bijection, injection and surjection|bijection]]''': a function that is injective and surjective. In other words, every element of the domain has ''exactly'' one image element and every element of the codomain has ''exactly'' one preimage element. For example, the green binary relation in the diagram is a bijection, but the red one is not.

{{Anchor|set-like-relation}}If relations over proper classes are allowed:
* '''Set-like''' (also called '''local'''): for all <math>x \in X</math>, the [[Class (set theory)|class]] of all <math>y \in Y</math> such that <math>yRx</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.<ref>{{cite book|title=Set theory: an introduction to independence proofs|page=102 |url=https://archive.org/details/settheoryintrodu0000kune/page/102/mode/2up|url-access=registration|last1=Kunen |first1=Kenneth|publisher=North-Holland|year=1980|isbn=0-444-85401-0|zbl=0443.03021}}</ref> The usual ordering < over the class of [[ordinal number]]s is a set-like relation, while its inverse > is not.{{citation needed|date=February 2022}}

== 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 <math>A</math>, that contains all the objects of interest, and work with the restriction <math>=_A</math> instead of <math>=</math>. Similarly, the "subset of" relation <math>\subseteq</math> needs to be restricted to have domain and codomain <math>P(A)</math> (the power set of a specific set <math>A</math>): the resulting set relation can be denoted by <math>\subseteq_A.</math> Also, the "member of" relation needs to be restricted to have domain <math>A</math> and codomain <math>P(A)</math> 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 [[Von Neumann–Bernays–Gödel set theory|NBG]] or [[Morse–Kelley set theory]], and allow the domain and codomain (and so the graph) to be [[proper class]]es: 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)</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.)<ref>{{cite book |title=A formalization of set theory without variables |last1=Tarski |first1=Alfred |author-link=Alfred Tarski |last2=Givant |first2=Steven |year=1987 |page=[https://archive.org/details/formalizationofs0000tars/page/3 3] |publisher=American Mathematical Society |isbn=0-8218-1041-3 |url=https://archive.org/details/formalizationofs0000tars/page/3 }}</ref> With this definition one can for instance define a binary relation over every set and its power set.

== Homogeneous relation ==
{{main|Homogeneous relation}}
A '''homogeneous relation'''<!---keep boldface: [[Homogeneous relation]] redirects to here---> over a set <math>X</math> is a binary relation over <math>X</math> and itself, i.e. it is a subset of the Cartesian product <math>X \times X.</math><ref name="Winter2007"/><ref name="Müller2012">{{cite book|author=M. E. Müller|title=Relational Knowledge Discovery|year=2012|publisher=Cambridge University Press|isbn=978-0-521-19021-3|page=22}}</ref><ref name="PahlDamrath2001-p496">{{cite book|author1=Peter J. Pahl|author2=Rudolf Damrath|title=Mathematical Foundations of Computational Engineering: A Handbook|year=2001|publisher=Springer Science & Business Media|isbn=978-3-540-67995-0|page=496}}</ref> It is also simply called a (binary) relation over <math>X</math>.

A homogeneous relation <math>R</math> over a set <math>X</math> may be identified with a [[Graph theory#Directed graph|directed simple graph permitting loops]], where <math>X</math> is the vertex set and <math>R</math> is the edge set (there is an edge from a vertex <math>x</math> to a vertex <math>y</math> if and only if <math>xRy</math>).
The set of all homogeneous relations <math>\mathcal{B}(X)</math> over a set <math>X</math> is the [[power set]] <math>2^{X \times X}</math> which is a [[Boolean algebra (structure)|Boolean algebra]] augmented with the [[Involution (mathematics)|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 <math>R</math> over a set <math>X</math> may have are:
* {{em|[[Reflexive relation|Reflexive]]}}: for all <math>x \in X,</math> <math>xRx</math>. For example, <math>\geq</math> is a reflexive relation but > is not.
* {{em|[[Irreflexive relation|Irreflexive]]}}: for all <math>x \in X,</math> not <math>xRx</math>. For example, <math>></math> is an irreflexive relation, but <math>\geq</math> is not.
* {{em|[[Symmetric relation|Symmetric]]}}: for all <math>x, y \in X,</math> if <math>xRy</math> then <math>yRx</math>. For example, "is a blood relative of" is a symmetric relation.
* {{em|[[Antisymmetric relation|Antisymmetric]]}}: for all <math>x, y \in X,</math> if <math>xRy</math> and <math>yRx</math> then <math>x = y.</math> For example, <math>\geq</math> is an antisymmetric relation.<ref>{{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}}</ref>
* {{em|[[Asymmetric relation|Asymmetric]]}}: for all <math>x, y \in X,</math> if <math>xRy</math> then not <math>yRx</math>. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.<ref>{{citation|first1=Yves|last1=Nievergelt|title=Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography|publisher=Springer-Verlag|year=2002|page=[https://books.google.com/books?id=_H_nJdagqL8C&pg=PA158 158]}}.</ref> For example, > is an asymmetric relation, but <math>\geq</math> is not.
* {{em|[[Transitive relation|Transitive]]}}: for all <math>x, y, z \in X,</math> if <math>xRy</math> and <math>yRz</math> then <math>xRz</math>. A transitive relation is irreflexive if and only if it is asymmetric.<ref>{{cite book|last1=Flaška|first1=V.|last2=Ježek|first2=J.|last3=Kepka|first3=T.|last4=Kortelainen|first4=J.|title=Transitive Closures of Binary Relations I|year=2007|publisher=School of Mathematics&nbsp;– Physics Charles University|location=Prague|page=1|url=http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf|url-status=dead|archive-url=https://web.archive.org/web/20131102214049/http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf|archive-date=2013-11-02}} Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".</ref> For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
* {{em|[[Connected relation|Connected]]}}: for all <math>x, y \in X,</math> if <math>x \neq y</math> then <math>xRy</math> or <math>yRx</math>.
* {{em|[[Connected relation|Strongly connected]]}}: for all <math>x, y \in X,</math> <math>xRy</math> or <math>yRx</math>.
* {{em|[[Dense order#Generalizations|Dense]]}}: 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 {{em|[[Partially ordered set#Formal definition|partial order]]}} is a relation that is reflexive, antisymmetric, and transitive. A {{em|[[Partially ordered set#Correspondence of strict and non-strict partial order relations|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.<ref>Joseph G. Rosenstein, ''Linear orderings'', Academic Press, 1982, {{ISBN|0-12-597680-1}}, p.&nbsp;4</ref> A {{em|[[Total order#Strict total order|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, "<math>x</math> divides <math>y</math>" is a partial, but not a total order on [[natural numbers]] <math>\N,</math> "<math>x < y</math>" is a strict total order on <math>\N,</math> and "<math>x</math> is parallel to <math>y</math>" 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 <math>X</math> may be subjected to closure operations like:
; {{em|[[Reflexive closure]]}}: the smallest reflexive relation over <math>X</math> containing <math>R</math>,
; {{em|[[Transitive closure]]}}: the smallest transitive relation over <math>X</math> containing <math>R</math>,
; {{em|[[Equivalence closure]]}}: the smallest [[equivalence relation]] over <math>X</math> containing <math>R</math>.

== 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 relation]]s. The inclusion <math>R \subseteq S,</math> meaning that <math>aRb</math> implies <math>aSb</math>, sets the scene in a [[Lattice (order theory)|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 [[composition of relations#Schröder rules|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 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 [[morphism]]s of this category are relations. The {{em|objects}} of the category [[Category of relations|Rel]] are sets, and the relation-morphisms compose as required in a [[Category (mathematics)|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 lattice]]s:
A '''concept''' <math>C \subset R</math> satisfies two properties:
* The [[logical matrix]] of <math>C</math> is the [[outer product]] of logical vectors <math>C_{i j} = u_i v_j , \quad u, v</math> [[logical vector]]s.{{clarify|reason=Given R, how are the logical vectors obtained?|date=June 2021}}
* <math>C</math> is maximal, not contained in any other outer product. Thus <math>C</math> 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".<ref>[[R. Berghammer]] & M. Winter (2013) "Decomposition of relations on concept lattices", [[Fundamenta Informaticae]] 126(1): 37–82 {{doi|10.3233/FI-2013-871}}</ref> The decomposition is
: <math>R = f E g^\textsf{T}</math>, where <math>f</math> and <math>g</math> are [[Function (mathematics)|function]]s, called {{em|mappings}} or left-total, functional relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial order <math>E</math> that belongs to the minimal decomposition <math>(f, g, E)</math> of the relation <math>R</math>."

Particular cases are considered below: <math>E</math> total order corresponds to Ferrers type, and <math>E</math> 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.<ref>[[Ki-Hang Kim]] (1982) ''Boolean Matrix Theory and Applications'', page 37, [[Marcel Dekker]] {{ISBN|0-8247-1788-0}}</ref> Structural analysis of relations with concepts provides an approach for [[data mining]].<ref>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}}</ref>

== Particular relations ==
* ''Proposition'': If <math>R</math> is a [[surjective relation]] and <math>R^\mathsf{T}</math> is its transpose, then <math>I \subseteq R^\textsf{T} R</math> where <math>I</math> is the <math>m \times m</math> identity relation.
* ''Proposition'': If <math>R</math> is a [[serial relation]], then <math>I \subseteq R R^\textsf{T}</math> where <math>I</math> is the <math>n \times n</math> 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 <math>Z = \{ x, y, z, \ldots \}</math> of [[Indicator (research)|indicator]]s. The partitioning relation <math>R = F G^\textsf{T}</math> is a [[composition of relations]] using {{em|functional}} relations <math>F \subseteq A \times Z \text{ and } G \subseteq B \times Z.</math> [[Jacques Riguet]] named these relations '''difunctional''' since the composition <math>F G^\mathsf{T}</math> involves functional relations, commonly called ''partial functions''.

In 1950 Riguet showed that such relations satisfy the inclusion:<ref>{{cite journal |last1=Riguet |first1=Jacques|author-link=Jacques Riguet|journal=Comptes rendus |date=January 1950 |url=https://gallica.bnf.fr/ark:/12148/bpt6k3182n/f2001.item |language=fr|title=Quelques proprietes des relations difonctionelles|volume=230|pages=1999–2000}}</ref>
: <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.<ref name="Büchi1989">{{cite book|author=Julius Richard Büchi|title=Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions|year=1989|publisher=Springer Science & Business Media|isbn=978-1-4613-8853-1|pages=35–37|author-link=Julius Richard Büchi}}</ref> More formally, a relation <math>R</math> 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 <math>X</math> and the <math>B_i</math> likewise a partition of a subset of <math>Y</math>.<ref>{{cite journal |last1=East |first1=James |last2=Vernitski |first2=Alexei |title=Ranks of ideals in inverse semigroups of difunctional binary relations |journal=Semigroup Forum |date=February 2018 |volume=96 |issue=1 |pages=21–30 |doi=10.1007/s00233-017-9846-9|arxiv=1612.04935|s2cid=54527913 }}</ref>

Using the notation <math>\{y \mid xRy\} = xR</math>, a difunctional relation can also be characterized as a relation <math>R</math> such that wherever <math>x_1 R</math> and <math>x_2 R</math> 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><ref name="BrinkKahl1997">{{cite book|author1=Chris Brink|author2=Wolfram Kahl|author3=Gunther Schmidt|title=Relational Methods in Computer Science|year=1997|publisher=Springer Science & Business Media|isbn=978-3-211-82971-4|page=200}}</ref>

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in [[database]] management."<ref>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}}</ref> Furthermore, difunctional relations are fundamental in the study of [[bisimulation]]s.<ref>{{Cite book | doi = 10.1007/978-3-662-44124-4_7
| chapter = Coalgebraic Simulations and Congruences| title = Coalgebraic Methods in Computer Science| volume = 8446| pages = 118| series = [[Lecture Notes in Computer Science]]| year = 2014| last1 = Gumm | first1 = H. P. | last2 = Zarrad | first2 = M. | isbn = 978-3-662-44123-7}}</ref>

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.<ref>J. Riguet (1951) "Les relations de Ferrers", [[Comptes Rendus]] 232: 1729,30</ref>

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.
<ref name="Schmidt p.77">{{cite book|last1=Schmidt|first1=Gunther|last2=Ströhlein|first2=Thomas|title=Relations and Graphs: Discrete Mathematics for Computer Scientists|url={{google books |plainurl=y |id=ZgarCAAAQBAJ|paged=277}}|date=2012|publisher=Springer Science & Business Media|isbn=978-3-642-77968-8|authorlink1=Gunther Schmidt |page=77}}</ref>

=== Contact ===
Suppose <math>B</math> is the [[power set]] of <math>A</math>, the set of all [[subset]]s of <math>A</math>. Then a relation <math>g</math> 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, <math>\epsilon = </math> "is an element of", satisfies these properties so <math>\epsilon</math> is a contact relation. The notion of a general contact relation was introduced by [[Georg Aumann]] in 1970.<ref>{{cite journal | url=https://www.zobodat.at/publikation_volumes.php?id=56359 | author=Georg Aumann | title=Kontakt-Relationen | journal=Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften München | volume=1970 | number=II | pages=67&ndash;77 | year=1971 }}</ref><ref>Anne K. Steiner (1970) [https://mathscinet.ams.org/mathscinet-getitem?mr=0309040 Review:''Kontakt-Relationen''] from [[Mathematical Reviews]]</ref>

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 (<math>\in</math>).<ref name=GS11/>{{rp|280}}

== Preorder R\R ==
Every relation <math>R</math> generates a [[preorder]] <math>R \backslash R</math> which is the [[Composition of relations#Quotients|left residual]].<ref>In this context, the symbol <math>\backslash</math> does not mean "[[set difference]]".</ref> 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^{\textsf{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 [[Transitive relation|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 (set theory)|inclusion]] relation &Omega; on the [[power set]] of <math>U</math> can be obtained in this way from the [[element (mathematics)|membership relation]] <math>\in</math> on subsets of <math>U</math>:
: <math>\Omega = \overline{\ni \bar{\in}} = \in \backslash \in .</math><ref name=GS11/>{{rp|283}}

== Fringe of a relation ==
Given a relation <math>R</math>, its '''fringe''' is the sub-relation defined as
<math display="block">\operatorname{fringe}(R) = R \cap \overline{R \bar{R}^\textsf{T} R}.</math>

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

On the other hand, <math>\operatorname{fringe}(R) = \emptyset</math> when <math>R</math> is a [[dense order|dense]], linear, strict order.<ref name=GS11>[[Gunther Schmidt]] (2011) ''Relational Mathematics'', pages 211−15, [[Cambridge University Press]] {{ISBN|978-0-521-76268-7}}</ref>

== Mathematical heaps ==
{{main|Heap (mathematics)}}
Given two sets <math>A</math> and <math>B</math>, 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 <math>b^\mathsf{T}</math> denotes the [[converse relation]] of <math>b</math>. In 1953 [[Viktor Wagner]] used properties of this ternary operation to define [[Semiheap|semiheaps]], heaps, and generalized heaps.<ref>[[Viktor Wagner]] (1953) "The theory of generalised heaps and generalised groups", [[Matematicheskii Sbornik]] 32(74): 545 to 632 {{mr|id=0059267}}</ref><ref>C.D. Hollings & M.V. Lawson (2017) ''Wagner's Theory of Generalised Heaps'', [[Springer books]] {{ISBN|978-3-319-63620-7}} {{mr|id=3729305}}</ref> 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 <math>A</math> and <math>B</math>, while the various types of semigroups appear in the case where <math>A = B</math>.
|author=Christopher Hollings
|title="Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups"<ref>Christopher Hollings (2014) ''Mathematics 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}}</ref>
}}

== 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 ==
* {{cite book |last=Schmidt |first=Gunther |author-link= |date=2010 |title=Relational Mathematics |url= |location=Berlin |publisher=Cambridge University Press |page= |isbn=9780511778810}}
* {{cite book|last1=Schmidt|first1=Gunther|last2=Ströhlein|first2=Thomas|title=Relations and Graphs: Discrete Mathematics for Computer Scientists|chapter-url={{google books |plainurl=y |id=ZgarCAAAQBAJ|paged=54}}|date=2012|chapter=Chapter 3: Heterogeneous relations|publisher=Springer Science & Business Media|isbn=978-3-642-77968-8|authorlink1=Gunther Schmidt}}
* [[Ernst Schröder (mathematician)|Ernst Schröder]] (1895) [https://archive.org/details/vorlesungenberd03mlgoog Algebra der Logik, Band III], via [[Internet Archive]]
* {{cite book |last=Codd |first=Edgar Frank |author-link=Edgar F. Codd |date=1990 |title=The Relational Model for Database Management: Version 2 |url=https://codeblab.com/wp-content/uploads/2009/12/rmdb-codd.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://codeblab.com/wp-content/uploads/2009/12/rmdb-codd.pdf |archive-date=2022-10-09 |url-status=live |location=Boston |publisher=[[Addison-Wesley]] |isbn=978-0201141924}}
* {{cite book |last=Enderton |first=Herbert |author-link=Herbert Enderton |date=1977 |title=Elements of Set Theory |location=Boston |publisher=[[Academic Press]] |isbn=978-0-12-238440-0}}
* {{cite book |last1=Kilp |first1=Mati |last2=Knauer |first2=Ulrich |last3=Mikhalev |first3=Alexander |date=2000 |title=Monoids, Acts and Categories: with Applications to Wreath Products and Graphs |location=Berlin |publisher=[[Walter de Gruyter|De Gruyter]] |isbn=978-3-11-015248-7}}
* {{cite book |last=Van Gasteren |first=Antonetta |author-link= |date=1990 |title=On the Shape of Mathematical Arguments |url= |location=Berlin |publisher=Springer |page= |isbn=9783540528494}}
* {{cite journal |last=Peirce |first=Charles Sanders |author-link=Charles Sanders Peirce |date=1873 |title=Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic |url= https://archive.org/details/descriptionanot00peirgoog/mode/2up |journal=Memoirs of the American Academy of Arts and Sciences |volume=9 |issue=2 |pages=317–178 |doi= 10.2307/25058006|jstor=25058006 |bibcode=1873MAAAS...9..317P |hdl=2027/hvd.32044019561034 |access-date=2020-05-05|hdl-access=free }}
* {{cite book |last=Schmidt |first=Gunther |author-link=Gunther Schmidt |date=2010 |title=Relational Mathematics |url=https://books.google.com/books?id=E4dREBTs5WsC |location=Cambridge |publisher=[[Cambridge University Press]] |isbn=978-0-521-76268-7}}

== External links ==
* {{springer|title=Binary relation|id=p/b016380}}

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