Homogeneous relation#Properties
{{Short description|Binary relation over a set and itself}}
In mathematics, a homogeneous relation (also called 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}}.{{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}}{{cite book|author=M. E. Müller|title=Relational Knowledge Discovery|year=2012|publisher=Cambridge University Press|isbn=978-0-521-19021-3|page=22}}{{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}} This is commonly phrased as "a relation on X"{{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}} or "a (binary) relation over X".{{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}}{{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}} An example of a homogeneous relation is the relation of kinship, where the relation is between people.
Common types of endorelations include orders, graphs, and 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 (x is R-related to y) 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
- : {{math|1=E = ∅}};
that is, {{math|x{{sub|1}}Ex{{sub|2}}}} holds never; - Universal relation
- : {{math|1=U = X × X}};
that is, {{math|x{{sub|1}}Ux{{sub|2}}}} holds always; - Identity relation (see also Identity function)
- : {{math|1=I = {(x, x) {{!}} x ∈ X}}};
that is, {{math|x{{sub|1}}Ix{{sub|2}}}} holds if and only if {{math|1=x{{sub|1}} = x{{sub|2}}}}.
= Example =
| class=wikitable style="float:right" | |||||||||||||||||
| || || Af || An || Ar || Au || Ca || Co || Eu || In || Ju || NA || Na || Pa || Ph || SA || Sc || So | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| African | Af | {{ya}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{na}} | {{ya}} |
| Antarctic | An | {{ya}} | {{ya}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{ya}} | {{ya}} | {{ya}} |
| Arabian | Ar | {{ya}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}} |
| Australian | Au | {{na}} | {{ya}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{ya}} |
| Caribbean | Ca | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{na}} | {{na}} |
| Cocos | Co | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{na}} |
| Eurasian | Eu | {{ya}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{na}} |
| Indian | In | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}} |
| Juan de Fuca | Ju | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{na}} |
| North american | NA | {{ya}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{ya}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{ya}} | {{ya}} | {{ya}} | {{na}} | {{na}} |
| Nazca | Na | {{na}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{ya}} | {{na}} | {{na}} |
| Pacific | Pa | {{na}} | {{ya}} | {{na}} | {{ya}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{ya}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} |
| Philippine | Ph | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} |
| South american | SA | {{ya}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} |
| Scotia | Sc | {{na}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{ya}} | {{na}} |
| Somali | So | {{ya}} | {{ya}} | {{ya}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{ya}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{na}} | {{ya}}
|+ Matrix representation of the relation "is adjacent to" on the set of tectonic plates |
| style="float:right;" |
| File:Tectonic plates (2022).svg are in contact is a homogenous relation, because both the first and second argument are from the same set, that is the set of tectonic plates on Earth.]] |
Sixteen large tectonic plates of the Earth's crust contact each other in a homogeneous relation. The relation can be expressed as a logical matrix with 1 (depicted "13px") indicating contact and 0 ("13px") 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}} : for all {{math|x ∈ X}}, {{math|xRx}}. For example, ≥ is a reflexive relation but > is not.
; {{em|Irreflexive}} (or {{em|strict}}) : for all {{math|x ∈ X}}, not {{math|xRx}}. For example, > is an irreflexive relation, but ≥ is not.
; {{em|Coreflexive}} : for all {{math|x, y ∈ X}}, if {{math|xRy}} then {{math|1=x = y}}.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). 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}} : 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 = x2}} 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}} : 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}} : 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 vacuously (the condition in the definition is always false).{{citation|first1=Douglas|last1=Smith|first2=Maurice|last2=Eggen|first3=Richard|last3=St. Andre|title=A Transition to Advanced Mathematics|edition=6th|publisher=Brooks/Cole|year=2006|isbn=0-534-39900-2|page=160}}
; {{em|Asymmetric}} : for all {{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.{{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]}}. 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}} : 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.{{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 – 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". 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-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-orders in constructive mathematics.
; {{em|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 orderings.
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}} : 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 orders.
; {{em|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|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|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.Since neither 5 divides 3, nor 3 divides 5, nor 3=5.
; {{em|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}} : every nonempty subset {{mvar|S}} of {{mvar|X}} contains a minimal element with respect to {{mvar|R}}. Well-foundedness implies the descending chain condition (that is, no infinite chain ... {{math|xnR...Rx3Rx2Rx1}} can exist). If the axiom of dependent choice is assumed, both conditions are equivalent.{{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 }}{{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}}
Moreover, all properties of binary relations in general also may apply to homogeneous relations:
; {{em|Set-like}} : for all {{math|x ∈ X}}, the 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}}.Gunther Schmidt & Thomas Strohlein (2012)[1987] {{Google books|ZgarCAAAQBAJ|Relations and Graphs|page=54}}
; {{em|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|total preorder}}, also called {{em|linear preorder}} or {{em|weak order}}, is a relation that is reflexive, transitive, and connected.
A {{em|partial order}}, also called {{em|order}},{{citation needed|date=March 2020}} is a relation that is reflexive, antisymmetric, and transitive. A {{em|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.Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, {{ISBN|0-12-597680-1}}, p. 4 A {{em|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.]] |
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= : Defined as {{math|1 = R= = {(x, x) {{!}} x ∈ X} ∪ R}} or the smallest reflexive relation over X containing R. This can be proven to be equal to the intersection of all reflexive relations containing R.
; {{em|Reflexive reduction}}, R≠ : Defined as {{math|1 = R≠ = R \ {(x, x) {{!}} x ∈ X}}} or the largest irreflexive relation over X contained in R.
; {{em|Transitive closure}}, R+ : 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+)=}}, the smallest preorder containing R.
; {{em|Reflexive transitive symmetric closure}}, R≡ : 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 |
| ! Reflexivity
! Symmetry ! Symbol ! Example |
|---|
| Directed graph
| | | | | → | |
| Undirected graph
| | {{yes|Symmetric}} | | | | |
| Dependency
| {{yes|Reflexive}} | {{yes|Symmetric}} | | | | |
| Tournament
| {{no|Irreflexive}} | {{no|Asymmetric}} | | | |
| Preorder
| {{yes|Reflexive}} | | {{yes|Transitive}} | | ≤ |
| Total preorder
| {{yes|Reflexive}} | | {{yes|Transitive}} | {{yes|Connected}} | ≤ | |
| Partial order
| {{yes|Reflexive}} | {{no|Antisymmetric}} | {{yes|Transitive}} | | ≤ | Subset |
| Strict partial order
| {{no|Irreflexive}} | {{no|Asymmetric}} | {{yes|Transitive}} | | < | Strict subset |
| Total order
| {{yes|Reflexive}} | {{no|Antisymmetric}} | {{yes|Transitive}} | {{yes|Connected}} | ≤ |
| 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 |
Enumeration
The set of all homogeneous relations over a set X is the set {{math|2X×X}}, which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on , it forms a monoid with involution where the identity element is the identity relation.{{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 }}
The number of distinct homogeneous relations over an n-element set is {{math|2n2}} {{OEIS|id=A002416}}:
{{number of relations}}
Notes:
- The number of irreflexive relations is the same as that of reflexive relations.
- The number of 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 partitions, which is the Bell number.
The homogeneous relations can be grouped into pairs (relation, complement), except that for {{math|1=n = 0}} the relation is its own complement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse complement).
Examples
- Order relations, including strict orders:
- Greater than
- Greater than or equal to
- Less than
- Less than or equal to
- Divides (evenly)
- Subset of
- Equivalence relations:
- Equality
- Parallel with (for affine spaces)
- Equinumerosity or "is in bijection with"
- Isomorphic
- 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 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 ⊆ X1 × ... × Xn}} for some natural number n and arbitrary sets X1, ..., Xn, it is also called an n-ary relation.
References
{{reflist}}