View source for Homogeneous relation

{{Short description|Binary relation over a set and itself}}
In [[mathematics]], a '''homogeneous relation'''<!---keep boldface: [[Homogeneous relation]] redirects to here---> (also called '''endorelation'''<!---[[endorelation]]--->) on a set ''X'' is a [[binary relation]] between ''X'' and itself, i.e. it is a subset of the [[Cartesian product]] {{math|''X'' × ''X''}}.<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><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> This is commonly phrased as "a relation on ''X''"<ref>{{cite book |last1=Mordeson |first1=John N. |last2=Nair |first2=Premchand S. |title=Fuzzy Mathematics: An Introduction for Engineers and Scientists |date=8 November 2012 |publisher=Physica |isbn=978-3-7908-1808-6 |page=2 |url=https://books.google.com/books?id=KfLwCAAAQBAJ&pg=PA2 |language=en}}</ref> or "a (binary) relation over ''X''".<ref>{{cite book |last1=Tanaev |first1=V. |last2=Gordon |first2=W. |last3=Shafransky |first3=Yakov M. |title=Scheduling Theory. Single-Stage Systems |date=6 December 2012 |publisher=Springer Science & Business Media |isbn=978-94-011-1190-4 |page=41 |url=https://books.google.com/books?id=ITDyCAAAQBAJ&pg=PA41 |language=en}}</ref><ref>{{cite book |last1=Meyer |first1=Bertrand |title=Touch of Class: Learning to Program Well with Objects and Contracts |date=29 June 2009 |publisher=Springer Science & Business Media |isbn=978-3-540-92145-5 |page=509 |url=https://books.google.com/books?id=AYNKAAAAQBAJ&pg=PA509 |language=en}}</ref> An example of a homogeneous relation is the relation of [[kinship]], where the relation is between people.

Common types of endorelations include [[order (mathematics)|order]]s, [[graph (discrete mathematics)|graph]]s, and [[equivalence relation|equivalences]]. Specialized studies of [[order theory]] and [[graph theory]] have developed understanding of endorelations. Terminology particular for graph theory is used for description, with an ordinary (undirected) graph presumed to correspond to a [[symmetric relation]], and a general endorelation corresponding to a [[directed graph]]. An endorelation ''R'' corresponds to a [[logical matrix]] of 0s and 1s, where the expression ''xRy'' corresponds to an edge between ''x'' and ''y'' in the graph, and to a 1 in the [[square matrix]] of ''R''. It is called an [[adjacency matrix]] in graph terminology.

== Particular homogeneous relations ==
Some particular homogeneous relations over a set ''X'' (with arbitrary elements {{math|''x''{{sub|1}}}}, {{math|''x''{{sub|2}}}}) are:
* '''Empty relation'''<!---[[empty relation]] redirects here---> 
*: {{math|1=''E'' = [[empty set|∅]]}};<br> that is, {{math|''x''{{sub|1}}''Ex''{{sub|2}}}} holds never;
*'''Universal relation'''<!---[[universal relation]]--->
*: {{math|1=''U'' = ''X'' × ''X''}};<br> that is, {{math|''x''{{sub|1}}''Ux''{{sub|2}}}} holds always;
*'''Identity relation'''<!---[[identity relation]]---> (see also ''[[Identity function]]'')
*: {{math|1=''I'' = {(''x'', ''x'') {{!}} ''x'' ∈ ''X''}}};<br> that is, {{math|''x''{{sub|1}}''Ix''{{sub|2}}}} holds if and only if {{math|1=''x''{{sub|1}} = ''x''{{sub|2}}}}.

=== Example ===
[[File:Plates_tect2_en.svg|right|thumb|300px]]
Fifteen large [[tectonic plate]]s of the Earth's crust contact each other in a homogeneous relation. The relation can be expressed as a [[logical matrix]] with 1 indicating contact and 0 no contact. This example expresses a symmetric relation.

== Properties ==
{{See also|:Category:Binary relations}}
Some important properties that a homogeneous relation {{mvar|R}} over a set {{mvar|X}} may have are:
; {{em|[[Reflexive relation|Reflexive]]}} : for all {{math|''x'' ∈ ''X''}}, {{math|''xRx''}}. For example, ≥ is a reflexive relation but > is not.
; {{em|[[Irreflexive relation|Irreflexive]]}} (or {{em|strict}}) : for all {{math|''x'' ∈ ''X''}}, not {{math|''xRx''}}. For example, > is an irreflexive relation, but ≥ is not.
; {{em|[[Coreflexive relation|Coreflexive]]}} : for all {{math|''x'', ''y'' ∈ ''X''}}, if {{math|''xRy''}} then {{math|1=''x'' = ''y''}}.<ref>Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). [https://www.researchgate.net/publication/221440123_Transposing_Relations_From_Maybe_Functions_to_Hash_Tables Transposing Relations: From Maybe Functions to Hash Tables]. In Mathematics of Program Construction (p. 337).</ref> For example, the relation over the integers in which each odd number is related to itself is a coreflexive relation. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation.
; {{em|[[Left quasi-reflexive]]}} : for all {{math|''x'', ''y'' ∈ ''X''}}, if {{math|''xRy''}} then {{math|''xRx''}}.
; {{em|[[Right quasi-reflexive]]}} : for all {{math|''x'', ''y'' ∈ ''X''}}, if {{math|''xRy''}} then {{math|''yRy''}}.
; {{em|[[Quasi-reflexive relation|Quasi-reflexive]]}} : for all {{math|''x'', ''y'' ∈ ''X''}}, if {{math|''xRy''}} then {{math|''xRx''}} and {{math|''yRy''}}. A relation is quasi-reflexive if, and only if, it is both left and right quasi-reflexive.

The previous 6 alternatives are far from being exhaustive; e.g., the binary relation {{math|''xRy''}} defined by {{math|1=''y'' = ''x''<sup>2</sup>}} is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair {{math|(0, 0)}}, and {{math|(2, 4)}}, but not {{math|(2, 2)}}, respectively. The latter two facts also rule out (any kind of) quasi-reflexivity.
; {{em|[[Symmetric relation|Symmetric]]}} : for all {{math|''x'', ''y'' ∈ ''X''}}, if {{math|''xRy''}} then {{math|''yRx''}}. For example, "is a blood relative of" is a symmetric relation, because {{mvar|x}} is a blood relative of {{mvar|y}} if and only if {{mvar|y}} is a blood relative of {{mvar|x}}.
; {{em|[[Antisymmetric relation|Antisymmetric]]}} : for all {{math|''x'', ''y'' ∈ ''X''}}, if {{math|''xRy''}} and {{math|''yRx''}} then {{math|1=''x'' = ''y''}}. For example, ≥ is an antisymmetric relation; so is >, but [[Vacuous truth|vacuously]] (the condition in the definition is always false).<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'' ∈ ''X''}}, if {{math|''xRy''}} then not {{math|''yRx''}}. 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 ≥ is not.

Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation {{math|''xRy''}} defined by {{math|''x'' > 2}} is neither symmetric nor antisymmetric, let alone asymmetric.
; {{em|[[Transitive relation|Transitive]]}} : for all {{math|''x'', ''y'', ''z'' ∈ ''X''}}, if {{math|''xRy''}} and {{math|''yRz''}} then {{math|''xRz''}}. 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|[[Antitransitive]]}} : for all {{math|''x'', ''y'', ''z'' ∈ ''X''}}, if {{math|''xRy''}} and {{math|''yRz''}} then never {{math|''xRz''}}.
; {{em|[[Co-transitivity|Co-transitive]]}} : if the complement of ''R'' is transitive. That is, for all {{math|''x'', ''y'', ''z'' ∈ ''X''}}, if {{math|''xRz''}}, then {{math|''xRy''}} or {{math|''yRz''}}. This is used in [[pseudo-order]]s in constructive mathematics.
; {{em|[[Quasitransitive relation|Quasitransitive]]}} : for all {{math|''x'', ''y'', ''z'' ∈ ''X''}}, if {{math|''xRy''}} and {{math|''yRz''}} but neither {{math|''yRx''}} nor {{math|''zRy''}}, then {{math|''xRz''}} but not {{math|''zRx''}}.  
; {{em|[[Transitivity of incomparability]]}} : for all {{math|''x'', ''y'', ''z'' ∈ ''X''}}, if {{mvar|x}} and {{mvar|y}} are incomparable with respect to {{mvar|R}} and if the same is true of {{mvar|y}} and {{mvar|z}}, then {{mvar|x}} and {{mvar|z}} are also incomparable with respect to {{mvar|R}}. This is used in [[weak ordering]]s.

Again, the previous 5 alternatives are not exhaustive. For example, the relation {{math|''xRy''}} if ({{math|1=''y'' = 0}} or {{math|1=''y'' = ''x''+1}}) satisfies none of these properties. On the other hand, the empty relation trivially satisfies all of them.
; {{em|[[Dense relation|Dense]]}} : for all {{math|''x'', ''y'' ∈ ''X''}} such that {{math|''xRy''}}, there exists some {{math|''z'' ∈ ''X''}} such that {{math|''xRz''}} and {{math|''zRy''}}. This is used in [[dense order]]s.
; {{em|[[Connected relation|Connected]]}} : for all {{math|''x'', ''y'' ∈ ''X''}}, if {{math|1=''x'' ≠ ''y''}} then {{math|''xRy''}} or {{math|''yRx''}}. This property is sometimes{{citation needed|reason=By whom?|date=June 2021}} called "total", which is distinct from the definitions of "left/right-total" given below.
; {{em|[[Connected relation|Strongly connected]]}} : for all {{math|''x'', ''y'' ∈ ''X''}}, {{math|''xRy''}} or {{math|''yRx''}}. This property, too, is sometimes{{citation needed|reason=By whom?|date=June 2021}} called "total", which is distinct from the definitions of "left/right-total" given below.
; {{em|[[Trichotomy (mathematics)|Trichotomous]]}} : for all {{math|''x'', ''y'' ∈ ''X''}}, exactly one of {{math|''xRy''}}, {{math|''yRx''}} or {{math|1=''x'' = ''y''}} holds. For example, > is a trichotomous relation on the real numbers, while the relation "divides" over the natural numbers is not.<ref>Since neither 5 divides 3, nor 3 divides 5, nor 3=5.</ref>
; {{em|[[Euclidean relation|Right Euclidean]]}} (or just {{em|Euclidean}}) : for all {{math|''x'', ''y'', ''z'' ∈ ''X''}}, if {{math|''xRy''}} and {{math|''xRz''}} then {{math|''yRz''}}. For example, = is a Euclidean relation because if {{math|1=''x'' = ''y''}} and {{math|1=''x'' = ''z''}} then {{math|1=''y'' = ''z''}}.
; {{em|Left Euclidean}} : for all {{math|''x'', ''y'', ''z'' ∈ ''X''}}, if {{math|''yRx''}} and {{math|''zRx''}} then {{math|''yRz''}}.
; {{em|[[Well-founded relation|Well-founded]]}} : every nonempty subset {{mvar|S}} of {{mvar|X}} contains a [[Maximal and minimal elements|minimal element]] with respect to {{mvar|R}}. Well-foundedness implies the [[descending chain condition]] (that is, no infinite chain ... {{math|''x''<sub>''n''</sub>''R''...''Rx''<sub>3</sub>''Rx''<sub>2</sub>''Rx''<sub>1</sub>}} can exist). If the [[axiom of dependent choice]] is assumed, both conditions are equivalent.<ref>{{cite web |title=Condition for Well-Foundedness |url=https://proofwiki.org/wiki/Condition_for_Well-Foundedness |website=ProofWiki |access-date=20 February 2019 |archive-date=20 February 2019 |archive-url=https://web.archive.org/web/20190220181521/https://proofwiki.org/wiki/Condition_for_Well-Foundedness |url-status=dead }}</ref><ref>{{cite book |last1=Fraisse |first1=R. |title=Theory of Relations, Volume 145 - 1st Edition |date=15 December 2000 |publisher=Elsevier |isbn=9780444505422 |page=46 |edition=1st |url=https://www.elsevier.com/books/theory-of-relations/fraisse/978-0-444-50542-2 |access-date=20 February 2019}}</ref>

Moreover, all properties of binary relations in general also may apply to homogeneous relations:
; {{em|Set-like}} : for all {{math|''x'' ∈ ''X''}}, the [[Class (set theory)|class]] of all {{mvar|y}} such that {{math|''yRx''}} is a set. (This makes sense only if relations over proper classes are allowed.)
; {{em|Left-unique}} : for all {{math|''x'', ''z'' ∈ ''X''}} and all {{math|''y'' ∈ ''Y''}}, if {{math|''xRy''}} and {{math|''zRy''}} then {{math|1=''x'' = ''z''}}. 
; {{em|Univalent}} : for all {{math|''x'' ∈ ''X''}} and all {{math|''y'', ''z'' ∈ ''Y''}}, if {{math|''xRy''}} and {{math|''xRz''}} then {{math|1=''y'' = ''z''}}.<ref>Gunther Schmidt & Thomas Strohlein (2012)[1987] {{Google books|ZgarCAAAQBAJ|Relations and Graphs|page=54}}</ref>
; {{em|[[total relation|Total]]}} (also called left-total) : for all {{math|''x'' ∈ ''X''}} there exists a {{math|''y'' ∈ ''Y''}} such that {{math|''xRy''}}. This property is different from the definition of ''connected'' (also called ''total'' by some authors).{{citation needed|date=June 2020}}
; {{em|Surjective}} (also called right-total) : for all {{math|''y'' ∈ ''Y''}}, there exists an {{math|''x'' ∈ ''X''}} such that ''xRy''.

A {{em|[[preorder]]}} is a relation that is reflexive and transitive. A {{em|[[Weak ordering#Total preorders|total preorder]]}}, also called {{em|linear preorder}} or {{em|weak order}}, is a relation that is reflexive, transitive, and connected.

A {{em|[[Partially ordered set#Formal definition|partial order]]}}, also called {{em|order}},{{citation needed|date=March 2020}} 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]]}}, also called {{em|strict order}},{{citation needed|date=March 2020}} is a relation that is irreflexive, antisymmetric, and transitive. A {{em|[[total order]]}}, also called {{em|linear order}}, {{em|simple order}}, or {{em|chain}}, 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]]}}, also called {{em|strict linear order}}, {{em|strict simple order}}, or {{em|strict chain}}, is a relation that is irreflexive, antisymmetric, transitive and connected.

A {{em|[[partial equivalence relation]]}} is a relation that is symmetric and transitive. An {{em|[[equivalence relation]]}} is a relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and total, since these properties imply reflexivity.

{| class="wikitable mw-collapsible mw-collapsed" style="float;"
|-
! Implications and conflicts between properties of homogeneous binary relations
|-
| [[File:BinRelProp Impl Confl.gif|thumb|750px|Implications (blue) and conflicts (red) between properties (yellow) of homogeneous binary relations. For example, every asymmetric relation is irreflexive (''"{{color|#808000|ASym}} {{color|#000080|⇒}} {{color|#808000|Irrefl}}"''), and no relation on a non-empty set can be both irreflexive and reflexive (''"{{color|#808000|Irrefl}} {{color|#800000|#}} {{color|#808000|Refl}}"''). Omitting the red edges results in a [[Hasse diagram]].]]
|}

== Operations ==
If ''R'' is a homogeneous relation over a set ''X'' then each of the following is a homogeneous relation over ''X'':
; {{em|[[Reflexive closure]]}}, ''R''<sup>=</sup> : Defined as {{math|1 = ''R''<sup>=</sup> = {(''x'', ''x'') {{!}} ''x'' ∈ ''X''} ∪ ''R''}} or the smallest reflexive relation over ''X'' containing ''R''. This can be proven to be equal to the [[Intersection (set theory)|intersection]] of all reflexive relations containing ''R''.
; {{em|Reflexive reduction}}, ''R''<sup>≠</sup> : Defined as {{math|1 = ''R''<sup>≠</sup> = ''R'' \ {(''x'', ''x'') {{!}} ''x'' ∈ ''X''}}} or the largest [[irreflexive]] relation over ''X'' contained in ''R''.
; {{em|[[Transitive closure]]}}, ''R''<sup>+</sup> : Defined as the smallest transitive relation over ''X'' containing ''R''. This can be seen to be equal to the intersection of all transitive relations containing ''R''.
; {{em|Reflexive transitive closure}}, ''R''* : Defined as {{math|1=''R''* = (''R''<sup>+</sup>)<sup>=</sup>}}, the smallest [[preorder]] containing ''R''.
; {{em|[[Reflexive transitive symmetric closure]]}}, ''R''<sup>≡</sup> : Defined as the smallest [[equivalence relation]] over ''X'' containing ''R''.

All operations defined in ''{{section link|Binary relation|Operations}}'' also apply to homogeneous relations.
 
: {| class="wikitable sortable" style="text-align:center;"
|+ Homogeneous relations by property
|-
!
! [[Reflexive relation|Reflexivity]]
! [[Symmetric relation|Symmetry]]
! [[Transitive relation|Transitivity]]
! [[Connected relation|Connectedness]]
! Symbol
! Example
|-
! [[Directed graph]]
|
|
|
|
| →
|
|-
! [[Undirected graph]]
|
| {{yes|Symmetric}}
|
|
|
|
|-
! [[Dependency relation|Dependency]]
| {{yes|Reflexive}}
| {{yes|Symmetric}}
|
|
|
|
|-
! [[Tournament (graph theory)|Tournament]]
| {{no|Irreflexive}}
| {{no|Asymmetric}}
|
|
|
| [[Pecking order]]
|-
! [[Preorder]]
| {{yes|Reflexive}}
|
| {{yes|Transitive}}
|
| ≤
| [[Preference]]
|-
! [[Weak ordering#Total preorders|Total preorder]]
| {{yes|Reflexive}}
|
| {{yes|Transitive}}
| {{yes|Connected}}
| ≤
|
|-
! [[Partially ordered set#Formal definition|Partial order]]
| {{yes|Reflexive}}
| {{no|Antisymmetric}}
| {{yes|Transitive}}
|
| ≤
| [[Subset]]
|-
! [[Partially ordered set#Correspondence of strict and non-strict partial order relations|Strict partial order]]
| {{no|Irreflexive}}
| {{no|Asymmetric}}
| {{yes|Transitive}}
|
| <
| Strict subset
|-
! [[Total order]]
| {{yes|Reflexive}}
| {{no|Antisymmetric}}
| {{yes|Transitive}}
| {{yes|Connected}}
| ≤
| [[Alphabetical order]]
|-
! [[Total order#Strict total order|Strict total order]]
| {{no|Irreflexive}}
| {{no|Asymmetric}}
| {{yes|Transitive}}
| {{yes|Connected}}
| <
| Strict alphabetical order
|-
! [[Partial equivalence relation]]
|
| {{yes|Symmetric}}
| {{yes|Transitive}}
|
|
|
|-
! [[Equivalence relation]]
| {{yes|Reflexive}}
| {{yes|Symmetric}}
| {{yes|Transitive}}
|
| ~, ≡
| [[Equality (mathematics)|Equality]]
|}

== Enumeration ==
The set of all homogeneous relations <math>\mathcal{B}(X)</math> over a set ''X'' is the set {{math|[[Power set|2<sup>''X''×''X''</sup>]]}}, 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 [[monoid with involution]] where the identity element is the identity relation.<ref>{{cite book |last1=Schmidt |first1=Gunther |last2=Ströhlein |first2=Thomas |title=Relations and Graphs: Discrete Mathematics for Computer Scientists |date=1993 |publisher=Springer |location=Berlin, Heidelberg |isbn=978-3-642-77968-8 |page=14 |url=https://link.springer.com/chapter/10.1007/978-3-642-77968-8_2|url-access=subscription |language=en |chapter=Homogeneous Relations|doi=10.1007/978-3-642-77968-8_2 }}</ref>

The number of distinct homogeneous relations over an ''n''-element set is {{math|2<sup>''n''<sup>2</sup></sup>}} {{OEIS|id=A002416}}:
{{number of relations}}

Notes:
* The number of irreflexive relations is the same as that of reflexive relations.
* The number of [[Partially ordered set#Correspondence of strict and non-strict partial order relations|strict partial orders]] (irreflexive transitive relations) is the same as that of partial orders.
* The number of strict weak orders is the same as that of total preorders.
* The total orders are the partial orders that are also total preorders. The number of preorders that are neither a partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders, minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.
* The number of equivalence relations is the number of [[Partition of a set|partition]]s, which is the [[Bell number]].

The homogeneous relations can be grouped into pairs (relation, [[Binary relation#Complement|complement]]), except that for {{math|1=''n'' = 0}} the relation is its own complement. The non-symmetric ones can be grouped into [[4-tuple|quadruple]]s (relation, complement, [[#Operations on binary relations|inverse]], inverse complement).

== Examples ==
* [[Order relation]]s, including [[strict order]]s:
** [[Greater than]]
** Greater than or equal to
** [[Less than]]
** Less than or equal to
** [[Divides]] (evenly)
** [[Subset]] of
* [[Equivalence relation]]s:
** [[Equality (mathematics)|Equality]]
** [[Parallel (geometry)|Parallel]] with (for [[affine space]]s)
** [[Equinumerosity]] or "is in [[bijection]] with"
** [[Isomorphism|Isomorphic]]
** [[Equipollence (geometry)|Equipollent line segments]]
* [[Tolerance relation]], a reflexive and symmetric relation:
** [[Dependency relation]], a finite tolerance relation
** [[Independency relation]], the complement of some dependency relation
* [[Kinship#Composition of relations|Kinship relations]]

== Generalizations ==
* A [[binary relation]] in general need not be homogeneous, it is defined to be a subset {{nowrap|''R'' ⊆ ''X'' × ''Y''}} for arbitrary sets ''X'' and ''Y''.
* A [[finitary relation]] is a subset {{nowrap|''R'' ⊆ ''X''<sub>1</sub> × ... × ''X''<sub>''n''</sub>}} for some [[natural number]] ''n'' and arbitrary sets ''X''<sub>1</sub>, ...,  ''X''<sub>''n''</sub>, it is also called an ''n''-ary relation.

== References ==
{{reflist}}

[[Category:Properties of binary relations]]