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Binary relation

From Wikipedia, the free encyclopedia
Relationship between elements of two sets
This article covers advanced notions. For basic topics, seeRelation (mathematics).
Transitive binary relations
SymmetricAntisymmetricConnectedWell-foundedHas joinsHas meetsReflexiveIrreflexiveAsymmetric
Total,
Semiconnex		 Anti-
reflexive
Equivalence relationGreen tickY✗✗✗✗✗Green tickY✗✗
Preorder(Quasiorder)✗✗✗✗✗✗Green tickY✗✗
Partial order✗Green tickY✗✗✗✗Green tickY✗✗
Total preorder✗✗Green tickY✗✗✗Green tickY✗✗
Total order✗Green tickYGreen tickY✗✗✗Green tickY✗✗
Prewellordering✗✗Green tickYGreen tickY✗✗Green tickY✗✗
Well-quasi-ordering✗✗✗Green tickY✗✗Green tickY✗✗
Well-ordering✗Green tickYGreen tickYGreen tickY✗✗Green tickY✗✗
Lattice✗Green tickY✗✗Green tickYGreen tickYGreen tickY✗✗
Join-semilattice✗Green tickY✗✗Green tickY✗Green tickY✗✗
Meet-semilattice✗Green tickY✗✗✗Green tickYGreen tickY✗✗
Strict partial order✗Green tickY✗✗✗✗✗Green tickYGreen tickY
Strict weak order✗Green tickY✗✗✗✗✗Green tickYGreen tickY
Strict total order✗Green tickYGreen tickY✗✗✗✗Green tickYGreen tickY
SymmetricAntisymmetricConnectedWell-foundedHas joinsHas meetsReflexiveIrreflexiveAsymmetric
Definitions,
for alla,b{\displaystyle a,b} andS≠∅:{\displaystyle S\neq \varnothing :}		aRb⇒bRa{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}aRb and bRa⇒a=b{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}a≠b⇒aRb or bRa{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}minSexists{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}a∨bexists{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}a∧bexists{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}aRa{\displaystyle aRa}not aRa{\displaystyle {\text{not }}aRa}aRb⇒not bRa{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
Green tickY indicates that the column's property is always true for the row's term (at the very left), while✗ indicates that the property is not guaranteed
in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,
is indicated byGreen tickY in the "Symmetric" column and✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require thehomogeneous relationR{\displaystyle R} betransitive: for alla,b,c,{\displaystyle a,b,c,} ifaRb{\displaystyle aRb} andbRc{\displaystyle bRc} thenaRc.{\displaystyle aRc.}
A term's definition may require additional properties that are not listed in this table.

An example of a binary relation R between two finite sets ofnatural numbers, A and B. Note that R is a subset of theCartesian product, A× B. In this example, R = {(a, b) ∈ A× B: a < b}.

Inmathematics, abinary relation associates some elements of oneset called thedomain with some elements of another set (possibly the same) called thecodomain.[1] Precisely, a binary relation over setsX{\displaystyle X} andY{\displaystyle Y} is a set ofordered pairs(x,y){\displaystyle (x,y)}, wherex{\displaystyle x} is an element ofX{\displaystyle X} andy{\displaystyle y} is an element ofY{\displaystyle Y}.[2] It encodes the common concept of relation: an elementx{\displaystyle x} isrelated to an elementy{\displaystyle y},if and only if the pair(x,y){\displaystyle (x,y)} belongs to the set of ordered pairs that defines the binary relation.

An example of a binary relation is the "divides" relation over the set ofprime numbersP{\displaystyle \mathbb {P} } and the set ofintegersZ{\displaystyle \mathbb {Z} }, in which each primep{\displaystyle p} is related to each integerz{\displaystyle z} that is amultiple ofp{\displaystyle p}, but not to an integer that is not amultiple ofp{\displaystyle p}. In this relation, for instance, the prime number2{\displaystyle 2} is related to numbers such as−4{\displaystyle -4},0{\displaystyle 0},6{\displaystyle 6},10{\displaystyle 10}, but not to1{\displaystyle 1} or9{\displaystyle 9}, just as the prime number3{\displaystyle 3} is related to0{\displaystyle 0},6{\displaystyle 6}, and9{\displaystyle 9}, but not to4{\displaystyle 4} or13{\displaystyle 13}.

A binary relation is called ahomogeneous relation whenX=Y{\displaystyle X=Y}. A binary relation is also called aheterogeneous relation when it is not necessary thatX=Y{\displaystyle X=Y}.

Binary relations, and especially homogeneous relations, are used in many branches of mathematics to model a wide variety of concepts. These include, among others:

Afunction may be defined as a binary relation that meets additional constraints.[3] Binary relations are also heavily used incomputer science.

A binary relation over setsX{\displaystyle X} andY{\displaystyle Y} can be identified with an element of thepower set of theCartesian productX×Y.{\displaystyle X\times Y.} Since a powerset is alattice forset inclusion (⊆{\displaystyle \subseteq }), relations can be manipulated using set operations (union,intersection, andcomplementation) andalgebra of sets.

In some systems ofaxiomatic set theory, relations are extended toclasses, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such asRussell's paradox.

A binary relation is the most studied special casen=2{\displaystyle n=2} of ann{\displaystyle n}-ary relation over setsX1,…,Xn{\displaystyle X_{1},\dots ,X_{n}}, which is a subset of theCartesian productX1×⋯×Xn.{\displaystyle X_{1}\times \cdots \times X_{n}.}[2]

Definition

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Given setsX{\displaystyle X} andY{\displaystyle Y}, the Cartesian productX×Y{\displaystyle X\times Y} is defined as{(x,y)∣x∈X and y∈Y},{\displaystyle \{(x,y)\mid x\in X{\text{ and }}y\in Y\},} and its elements are calledordered pairs.

Abinary relationR{\displaystyle R} over setsX{\displaystyle X} andY{\displaystyle Y} is a subset ofX×Y.{\displaystyle X\times Y.}[2][4] The setX{\displaystyle X} is called thedomain[2] orset of departure ofR{\displaystyle R}, and the setY{\displaystyle Y} thecodomain orset of destination ofR{\displaystyle R}. In order to specify the choices of the setsX{\displaystyle X} andY{\displaystyle Y}, some authors define abinary relation orcorrespondence as an ordered triple(X,Y,G){\displaystyle (X,Y,G)}, whereG{\displaystyle G} is a subset ofX×Y{\displaystyle X\times Y} called thegraph of the binary relation. The statement(x,y)∈R{\displaystyle (x,y)\in R} reads "x{\displaystyle x} isR{\displaystyle R}-related toy{\displaystyle y}" and is denoted byxRy{\displaystyle xRy}.[5][6][7][a] Thedomain of definition oractive domain[2] ofR{\displaystyle R} is the set of allx{\displaystyle x} such thatxRy{\displaystyle xRy} for at least oney{\displaystyle y}. Thecodomain of definition,active codomain,[2]image orrange ofR{\displaystyle R} is the set of ally{\displaystyle y} such thatxRy{\displaystyle xRy} for at least onex{\displaystyle x}. Thefield ofR{\displaystyle R} is the union of its domain of definition and its codomain of definition.[9][10][11]

WhenX=Y,{\displaystyle X=Y,} a binary relation is called ahomogeneous relation (orendorelation). To emphasize the fact thatX{\displaystyle X} andY{\displaystyle Y} are allowed to be different, a binary relation is also called aheterogeneous relation.[12][13][14] The prefixhetero is from the Greek ἕτερος (heteros, "other, another, different").

A heterogeneous relation has been called arectangular relation,[14] suggesting that it does not have the square-like symmetry of ahomogeneous relation on a set whereA=B.{\displaystyle A=B.} Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "... a variant of the theory has evolved that treats relations from the very beginning asheterogeneous orrectangular, i.e. as relations where the normal case is that they are relations between different sets."[15]

The termscorrespondence,[16]dyadic relation andtwo-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian productX×Y{\displaystyle X\times Y} without reference toX{\displaystyle X} andY{\displaystyle Y}, and reserve the term "correspondence" for a binary relation with reference toX{\displaystyle X} andY{\displaystyle Y}.[citation needed]

In a binary relation, the order of the elements is important; ifx≠y{\displaystyle x\neq y} thenyRx{\displaystyle yRx} can be true or false independently ofxRy{\displaystyle xRy}. For example,3{\displaystyle 3} divides9{\displaystyle 9}, but9{\displaystyle 9} does not divide3{\displaystyle 3}.

Operations

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Union

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IfR{\displaystyle R} andS{\displaystyle S} are binary relations over setsX{\displaystyle X} andY{\displaystyle Y} thenR∪S={(x,y)∣xRy or xSy}{\displaystyle R\cup S=\{(x,y)\mid xRy{\text{ or }}xSy\}} is theunion relation ofR{\displaystyle R} andS{\displaystyle S} overX{\displaystyle X} andY{\displaystyle Y}.

The identity element is the empty relation. For example,≤{\displaystyle \leq } is the union of < and =, and≥{\displaystyle \geq } is the union of > and =.

Intersection

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IfR{\displaystyle R} andS{\displaystyle S} are binary relations over setsX{\displaystyle X} andY{\displaystyle Y} thenR∩S={(x,y)∣xRy and xSy}{\displaystyle R\cap S=\{(x,y)\mid xRy{\text{ and }}xSy\}} is theintersection relation ofR{\displaystyle R} andS{\displaystyle S} overX{\displaystyle X} andY{\displaystyle Y}.

The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2".

Composition

[edit]
Main article:Composition of relations

IfR{\displaystyle R} is a binary relation over setsX{\displaystyle X} andY{\displaystyle Y}, andS{\displaystyle S} is a binary relation over setsY{\displaystyle Y} andZ{\displaystyle Z} thenS∘R={(x,z)∣ there exists y∈Y such that xRy and ySz}{\displaystyle S\circ R=\{(x,z)\mid {\text{ there exists }}y\in Y{\text{ such that }}xRy{\text{ and }}ySz\}} (also denoted byR;S{\displaystyle R;S}) is thecomposition relation ofR{\displaystyle R} andS{\displaystyle S} overX{\displaystyle X} andZ{\displaystyle Z}.

The identity element is the identity relation. The order ofR{\displaystyle R} andS{\displaystyle S} in the notationS∘R,{\displaystyle S\circ R,} used here agrees with the standard notational order forcomposition of functions. For example, the composition (is parent of)∘{\displaystyle \circ }(is mother of) yields (is maternal grandparent of), while the composition (is mother of)∘{\displaystyle \circ }(is parent of) yields (is grandmother of). For the former case, ifx{\displaystyle x} is the parent ofy{\displaystyle y} andy{\displaystyle y} is the mother ofz{\displaystyle z}, thenx{\displaystyle x} is the maternal grandparent ofz{\displaystyle z}.

Converse

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Main article:Converse relation
See also:Duality (order theory)

IfR{\displaystyle R} is a binary relation over setsX{\displaystyle X} andY{\displaystyle Y} thenRT={(y,x)∣xRy}{\displaystyle R^{\textsf {T}}=\{(y,x)\mid xRy\}} is theconverse relation,[17] also calledinverse relation,[18] ofR{\displaystyle R} overY{\displaystyle Y} andX{\displaystyle X}.

For example,={\displaystyle =} is the converse of itself, as is≠{\displaystyle \neq }, and<{\displaystyle <} and>{\displaystyle >} are each other's converse, as are≤{\displaystyle \leq } and≥.{\displaystyle \geq .} A binary relation is equal to its converse if and only if it issymmetric.

Complement

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Main article:Complementary relation

IfR{\displaystyle R} is a binary relation over setsX{\displaystyle X} andY{\displaystyle Y} thenR¯={(x,y)∣¬xRy}{\displaystyle {\bar {R}}=\{(x,y)\mid \neg xRy\}} (also denoted by¬R{\displaystyle \neg R}) is thecomplementary relation ofR{\displaystyle R} overX{\displaystyle X} andY{\displaystyle Y}.

For example,={\displaystyle =} and≠{\displaystyle \neq } are each other's complement, as are⊆{\displaystyle \subseteq } and⊈{\displaystyle \not \subseteq },⊇{\displaystyle \supseteq } and⊉{\displaystyle \not \supseteq },∈{\displaystyle \in } and∉{\displaystyle \not \in }, and fortotal orders also<{\displaystyle <} and≥{\displaystyle \geq }, and>{\displaystyle >} and≤{\displaystyle \leq }.

The complement of theconverse relationRT{\displaystyle R^{\textsf {T}}} is the converse of the complement:RT¯=R¯T.{\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.}

IfX=Y,{\displaystyle X=Y,} the complement has the following properties:

  • If a relation is symmetric, then so is the complement.
  • The complement of a reflexive relation is irreflexive—and vice versa.
  • The complement of astrict weak order is a total preorder—and vice versa.

Restriction

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Main article:Restriction (mathematics)

IfR{\displaystyle R} is a binaryhomogeneous relation over a setX{\displaystyle X} andS{\displaystyle S} is a subset ofX{\displaystyle X} thenR|S={(x,y)∣xRy and x∈S and y∈S}{\displaystyle R_{\vert S}=\{(x,y)\mid xRy{\text{ and }}x\in S{\text{ and }}y\in S\}} is therestriction relation ofR{\displaystyle R} toS{\displaystyle S} overX{\displaystyle X}.

IfR{\displaystyle R} is a binary relation over setsX{\displaystyle X} andY{\displaystyle Y} and ifS{\displaystyle S} is a subset ofX{\displaystyle X} thenR|S={(x,y)∣xRy and x∈S}{\displaystyle R_{\vert S}=\{(x,y)\mid xRy{\text{ and }}x\in S\}} is theleft-restriction relation ofR{\displaystyle R} toS{\displaystyle S} overX{\displaystyle X} andY{\displaystyle Y}.[clarification needed]

If a relation isreflexive, irreflexive,symmetric,antisymmetric,asymmetric,transitive,total,trichotomous, apartial order,total order,strict weak order,total preorder (weak order), or anequivalence relation, then so too are its restrictions.

However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "x{\displaystyle x} is parent ofy{\displaystyle y}" to females yields the relation "x{\displaystyle x} is mother of the womany{\displaystyle y}"; its transitive closure does not relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.

Also, the various concepts ofcompleteness (not to be confused with being "total") do not carry over to restrictions. For example, over thereal numbers a property of the relation≤{\displaystyle \leq } is that everynon-empty subsetS⊆R{\displaystyle S\subseteq \mathbb {R} } with anupper bound inR{\displaystyle \mathbb {R} } has aleast upper bound (also called supremum) inR.{\displaystyle \mathbb {R} .} However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation≤{\displaystyle \leq } to the rational numbers.

A binary relationR{\displaystyle R} over setsX{\displaystyle X} andY{\displaystyle Y} is said to becontained in a relationS{\displaystyle S} overX{\displaystyle X} andY{\displaystyle Y}, writtenR⊆S,{\displaystyle R\subseteq S,} ifR{\displaystyle R} is a subset ofS{\displaystyle S}, that is, for allx∈X{\displaystyle x\in X} andy∈Y,{\displaystyle y\in Y,} ifxRy{\displaystyle xRy}, thenxSy{\displaystyle xSy}. IfR{\displaystyle R} is contained inS{\displaystyle S} andS{\displaystyle S} is contained inR{\displaystyle R}, thenR{\displaystyle R} andS{\displaystyle S} are calledequal writtenR=S{\displaystyle R=S}. IfR{\displaystyle R} is contained inS{\displaystyle S} butS{\displaystyle S} is not contained inR{\displaystyle R}, thenR{\displaystyle R} is said to besmaller thanS{\displaystyle S}, writtenR⊊S.{\displaystyle R\subsetneq S.} For example, on therational numbers, the relation>{\displaystyle >} is smaller than≥{\displaystyle \geq }, and equal to the composition>∘>{\displaystyle >\circ >}.

Matrix representation

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Binary relations over setsX{\displaystyle X} andY{\displaystyle Y} can be represented algebraically bylogical matrices indexed byX{\displaystyle X} andY{\displaystyle Y} with entries in theBoolean semiring (addition corresponds to OR and multiplication to AND) wherematrix addition corresponds to union of relations,matrix multiplication corresponds to composition of relations (of a relation overX{\displaystyle X} andY{\displaystyle Y} and a relation overY{\displaystyle Y} andZ{\displaystyle Z}),[19] theHadamard product corresponds to intersection of relations, thezero matrix corresponds to the empty relation, and thematrix of ones corresponds to the universal relation. Homogeneous relations (whenX=Y{\displaystyle X=Y}) form amatrix semiring (indeed, amatrix semialgebra over the Boolean semiring) where theidentity matrix corresponds to the identity relation.[20]

Examples

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2nd example relation
ballcardollcup
John+−−−
Mary−−+−
Venus−+−−
1st example relation
ballcardollcup
John+−−−
Mary−−+−
Ian−−−−
Venus−+−−
  1. The following example shows that the choice of codomain is important. Suppose there are four objectsA={ball, car, doll, cup}{\displaystyle A=\{{\text{ball, car, doll, cup}}\}} and four peopleB={John, Mary, Ian, Venus}.{\displaystyle B=\{{\text{John, Mary, Ian, Venus}}\}.} A possible relation onA{\displaystyle A} andB{\displaystyle B} is the relation "is owned by", given byR={(ball, John),(doll, Mary),(car, Venus)}.{\displaystyle R=\{({\text{ball, John}}),({\text{doll, Mary}}),({\text{car, Venus}})\}.} That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing; see the 1st example. As a set,R{\displaystyle R} does not involve Ian, and thereforeR{\displaystyle R} could have been viewed as a subset ofA×{John, Mary, Venus},{\displaystyle A\times \{{\text{John, Mary, Venus}}\},} i.e. a relation overA{\displaystyle A} and{John, Mary, Venus};{\displaystyle \{{\text{John, Mary, Venus}}\};} see the 2nd example. But in that second example,R{\displaystyle R} contains no information about the ownership by Ian.

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

    Oceans and continents (islands omitted)
    Ocean borders continent
    NASAAFEUASAUAA
    Indian0010111
    Arctic1001100
    Atlantic1111001
    Pacific1100111
  2. LetA={Indian,Arctic,Atlantic,Pacific}{\displaystyle A=\{{\text{Indian}},{\text{Arctic}},{\text{Atlantic}},{\text{Pacific}}\}}, theoceans of the globe, andB={NA,SA,AF,EU,AS,AU,AA}{\displaystyle B=\{{\text{NA}},{\text{SA}},{\text{AF}},{\text{EU}},{\text{AS}},{\text{AU}},{\text{AA}}\}}, thecontinents. LetaRb{\displaystyle aRb} represent that oceana{\displaystyle a} borders continentb{\displaystyle b}. Then thelogical matrix for this relation is:
    R=(0010111100110011110011100111).{\displaystyle R={\begin{pmatrix}0&0&1&0&1&1&1\\1&0&0&1&1&0&0\\1&1&1&1&0&0&1\\1&1&0&0&1&1&1\end{pmatrix}}.}
    The connectivity of the planet Earth can be viewed throughRRT{\displaystyle RR^{\mathsf {T}}} andRTR{\displaystyle R^{\mathsf {T}}R}, the former being a4×4{\displaystyle 4\times 4} relation onA{\displaystyle A}, which is the universal relation (A×A{\displaystyle A\times A} or a logical matrix of all ones). This universal relation reflects the fact that every ocean is separated from the others by at most one continent. On the other hand,RTR{\displaystyle R^{\mathsf {T}}R} is a relation onB×B{\displaystyle B\times B} whichfails to be universal because at least two oceans must be traversed to voyage fromEurope toAustralia.
  3. Visualization of relations leans ongraph theory: For relations on a set (homogeneous relations), adirected graph illustrates a relation and agraph asymmetric relation. For heterogeneous relations ahypergraph has edges possibly with more than two nodes, and can be illustrated by abipartite graph.Just as theclique is integral to relations on a set, sobicliques are used to describe heterogeneous relations; indeed, they are the "concepts" that generate a lattice associated with a relation.
    The varioust{\displaystyle t} axes represent time for observers in motion, the correspondingx{\displaystyle x} axes are their lines of simultaneity.
  4. Hyperbolic orthogonality: Time and space are different categories, and temporal properties are separate from spatial properties. The idea ofsimultaneous events is simple inabsolute space and time since each timet{\displaystyle t} determines a simultaneoushyperplane in that cosmology.Hermann Minkowski changed that when he articulated the notion ofrelative simultaneity, which exists when spatial events are "normal" to a time characterized by a velocity. He used an indefinite inner product, and specified that a time vector is normal to a space vector when that product is zero. The indefinite inner product in acomposition algebra is given by
    ⟨x,z⟩=xz¯+x¯z{\displaystyle \langle x,z\rangle =x{\bar {z}}+{\bar {x}}z\;} where the overbar denotes conjugation.
    As a relation between some temporal events and some spatial events,hyperbolic orthogonality (as found insplit-complex numbers) is a heterogeneous relation.[21]
  5. Ageometric configuration can be considered a relation between its points and its lines. The relation is expressed asincidence. Finite and infinite projective and affine planes are included.Jakob Steiner pioneered the cataloguing of configurations with theSteiner systemsS⁡(t,k,n){\displaystyle \operatorname {S} (t,k,n)} which have an n-element setS{\displaystyle \operatorname {S} } and a set of k-element subsets calledblocks, such that a subset witht{\displaystyle t} elements lies in just one block. Theseincidence structures have been generalized withblock designs. Theincidence matrix used in these geometrical contexts corresponds to the logical matrix used generally with binary relations.
    An incidence structure is a tripleD=(V,B,I){\displaystyle \mathbf {D} =(V,\mathbf {B} ,I)} whereV{\displaystyle V} andB{\displaystyle \mathbf {B} } are any two disjoint sets andI{\displaystyle I} is a binary relation betweenV{\displaystyle V} andB{\displaystyle \mathbf {B} }, i.e.I⊆V×B.{\displaystyle I\subseteq V\times \mathbf {B} .} The elements ofV{\displaystyle V} will be calledpoints, those ofB{\displaystyle \mathbf {B} }blocks, and those ofI{\displaystyle I}flags.[22]

Types of binary relations

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Examples of four types of binary relations over thereal numbers: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black).

Some important types of binary relationsR{\displaystyle R} over setsX{\displaystyle X} andY{\displaystyle Y} are listed below.

Uniqueness properties:

Totality properties (only definable if the domainX{\displaystyle X} and codomainY{\displaystyle Y} are specified):

Uniqueness and totality properties (only definable if the domainX{\displaystyle X} and codomainY{\displaystyle Y} are specified):

  • Afunction (also calledmapping[24]): a binary relation that is functional and total. In other words, every element of the domain hasexactly one image element. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
  • Aninjection: a function that is injective. For example, the green relation in the diagram is an injection, but the red one is not; the black and the blue relation is not even a function.
  • Asurjection: a function that is surjective. For example, the green relation in the diagram is a surjection, but the red one is not.
  • Abijection: a function that is injective and surjective. In other words, every element of the domain hasexactly one image element and every element of the codomain hasexactly one preimage element. For example, the green binary relation in the diagram is a bijection, but the red one is not.

If relations over proper classes are allowed:

Sets versus classes

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Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, to model the general concept of "equality" as a binary relation={\displaystyle =}, 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" setA{\displaystyle A}, that contains all the objects of interest, and work with the restriction=A{\displaystyle =_{A}} instead of={\displaystyle =}. Similarly, the "subset of" relation⊆{\displaystyle \subseteq } needs to be restricted to have domain and codomainP(A){\displaystyle P(A)} (the power set of a specific setA{\displaystyle A}): the resulting set relation can be denoted by⊆A.{\displaystyle \subseteq _{A}.} Also, the "member of" relation needs to be restricted to have domainA{\displaystyle A} and codomainP(A){\displaystyle P(A)} to obtain a binary relation∈A{\displaystyle \in _{A}} that is a set.Bertrand Russell has shown that assuming∈{\displaystyle \in } to be defined over all sets leads to a contradiction innaive set theory, seeRussell's paradox.

Another solution to this problem is to use a set theory with proper classes, such asNBG orMorse–Kelley set theory, and allow the domain and codomain (and so the graph) to beproper classes: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple(X,Y,G){\displaystyle (X,Y,G)}, as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.)[31] With this definition one can for instance define a binary relation over every set and its power set.

Homogeneous relation

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Main article:Homogeneous relation

Ahomogeneous relation over a setX{\displaystyle X} is a binary relation overX{\displaystyle X} and itself, i.e. it is a subset of the Cartesian productX×X.{\displaystyle X\times X.}[14][32][33] It is also simply called a (binary) relation overX{\displaystyle X}.

A homogeneous relationR{\displaystyle R} over a setX{\displaystyle X} may be identified with adirected simple graph permitting loops, whereX{\displaystyle X} is the vertex set andR{\displaystyle R} is the edge set (there is an edge from a vertexx{\displaystyle x} to a vertexy{\displaystyle y} if and only ifxRy{\displaystyle xRy}).The set of all homogeneous relationsB(X){\displaystyle {\mathcal {B}}(X)} over a setX{\displaystyle X} is thepower set2X×X{\displaystyle 2^{X\times X}} which is aBoolean algebra augmented with theinvolution of mapping of a relation to itsconverse relation. Consideringcomposition of relations as abinary operation onB(X){\displaystyle {\mathcal {B}}(X)}, it forms asemigroup with involution.

Some important properties that a homogeneous relationR{\displaystyle R} over a setX{\displaystyle X} may have are:

Apartial order is a relation that is reflexive, antisymmetric, and transitive. Astrict partial order is a relation that is irreflexive, asymmetric, and transitive. Atotal order is a relation that is reflexive, antisymmetric, transitive and connected.[37] Astrict total order is a relation that is irreflexive, asymmetric, transitive and connected.Anequivalence relation is a relation that is reflexive, symmetric, and transitive.For example, "x{\displaystyle x} dividesy{\displaystyle y}" is a partial, but not a total order onnatural numbersN,{\displaystyle \mathbb {N} ,} "x<y{\displaystyle x<y}" is a strict total order onN,{\displaystyle \mathbb {N} ,} and "x{\displaystyle x} is parallel toy{\displaystyle y}" is an equivalence relation on the set of all lines in theEuclidean plane.

All operations defined in section§ Operations also apply to homogeneous relations.Beyond that, a homogeneous relation over a setX{\displaystyle X} may be subjected to closure operations like:

Reflexive closure
the smallest reflexive relation overX{\displaystyle X} containingR{\displaystyle R},
Transitive closure
the smallest transitive relation overX{\displaystyle X} containingR{\displaystyle R},
Equivalence closure
the smallestequivalence relation overX{\displaystyle X} containingR{\displaystyle R}.

Calculus of relations

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Developments inalgebraic logic have facilitated usage of binary relations. Thecalculus of relations includes thealgebra of sets, extended bycomposition of relations and the use ofconverse relations. The inclusionR⊆S,{\displaystyle R\subseteq S,} meaning thataRb{\displaystyle aRb} impliesaSb{\displaystyle aSb}, sets the scene in alattice of relations. But sinceP⊆Q≡(P∩Q¯=∅)≡(P∩Q=P),{\displaystyle P\subseteq Q\equiv (P\cap {\bar {Q}}=\varnothing )\equiv (P\cap Q=P),} the inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according toSchröder rules, provides a calculus to work in thepower set ofA×B.{\displaystyle A\times B.}

In contrast to homogeneous relations, thecomposition of relations operation is only apartial function. The necessity of matching target to source of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter ofcategory theory as in thecategory of sets, except that themorphisms of this category are relations. Theobjects of the categoryRel are sets, and the relation-morphisms compose as required in acategory.[citation needed]

Induced concept lattice

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Binary relations have been described through their inducedconcept lattices:AconceptC⊂R{\displaystyle C\subset R} satisfies two properties:

For a given relationR⊆X×Y,{\displaystyle R\subseteq X\times Y,} the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion⊑{\displaystyle \sqsubseteq } forming apreorder.

TheMacNeille completion theorem (1937) (that any partial order may be embedded in acomplete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices".[38] The decomposition is

R=fEgT{\displaystyle R=fEg^{\textsf {T}}}, wheref{\displaystyle f} andg{\displaystyle g} arefunctions, calledmappings or left-total, functional relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial orderE{\displaystyle E} that belongs to the minimal decomposition(f,g,E){\displaystyle (f,g,E)} of the relationR{\displaystyle R}."

Particular cases are considered below:E{\displaystyle E} total order corresponds to Ferrers type, andE{\displaystyle E} identity corresponds to difunctional, a generalization ofequivalence relation on a set.

Relations may be ranked by theSchein rank which counts the number of concepts necessary to cover a relation.[39] Structural analysis of relations with concepts provides an approach fordata mining.[40]

Particular relations

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Difunctional

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The idea of a difunctional relation is to partition objects by distinguishing attributes, as a generalization of the concept of anequivalence relation. One way this can be done is with an intervening setZ={x,y,z,…}{\displaystyle Z=\{x,y,z,\ldots \}} ofindicators. The partitioning relationR=FGT{\displaystyle R=FG^{\textsf {T}}} is acomposition of relations usingfunctional relationsF⊆A×Z and G⊆B×Z.{\displaystyle F\subseteq A\times Z{\text{ and }}G\subseteq B\times Z.}Jacques Riguet named these relationsdifunctional since the compositionFGT{\displaystyle FG^{\mathsf {T}}} involves functional relations, commonly calledpartial functions.

In 1950 Riguet showed that such relations satisfy the inclusion:[41]

RRTR⊆R{\displaystyle RR^{\textsf {T}}R\subseteq R}

Inautomata theory, the termrectangular relation has also been used to denote a difunctional relation. This terminology recalls the fact that, when represented as alogical matrix, the columns and rows of a difunctional relation can be arranged as ablock matrix with rectangular blocks of ones on the (asymmetric) main diagonal.[42] More formally, a relationR{\displaystyle R} onX×Y{\displaystyle X\times Y} is difunctional if and only if it can be written as the union of Cartesian productsAi×Bi{\displaystyle A_{i}\times B_{i}}, where theAi{\displaystyle A_{i}} are a partition of a subset ofX{\displaystyle X} and theBi{\displaystyle B_{i}} likewise a partition of a subset ofY{\displaystyle Y}.[43]

Using the notation{y∣xRy}=xR{\displaystyle \{y\mid xRy\}=xR}, a difunctional relation can also be characterized as a relationR{\displaystyle R} such that whereverx1R{\displaystyle x_{1}R} andx2R{\displaystyle x_{2}R} have a non-empty intersection, then these two sets coincide; formallyx1∩x2≠∅{\displaystyle x_{1}\cap x_{2}\neq \varnothing } impliesx1R=x2R.{\displaystyle x_{1}R=x_{2}R.}[44]

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies indatabase management."[45] Furthermore, difunctional relations are fundamental in the study ofbisimulations.[46]

In the context of homogeneous relations, apartial equivalence relation is difunctional.

Ferrers type

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Astrict order on a set is a homogeneous relation arising inorder theory.In 1951Jacques Riguet adopted the ordering of aninteger partition, called aFerrers diagram, to extend ordering to binary relations in general.[47]

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 isRR¯TR⊆R.{\displaystyle R{\bar {R}}^{\textsf {T}}R\subseteq R.}

If any one of the relationsR,R¯,RT{\displaystyle R,{\bar {R}},R^{\textsf {T}}} is of Ferrers type, then all of them are.[48]

Contact

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SupposeB{\displaystyle B} is thepower set ofA{\displaystyle A}, the set of allsubsets ofA{\displaystyle A}. Then a relationg{\displaystyle g} is acontact relation if it satisfies three properties:

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

Theset membership relation,ϵ={\displaystyle \epsilon =} "is an element of", satisfies these properties soϵ{\displaystyle \epsilon } is a contact relation. The notion of a general contact relation was introduced byGeorg Aumann in 1970.[49][50]

In terms of the calculus of relations, sufficient conditions for a contact relation includeCTC¯⊆∋C¯≡C∋C¯¯⊆C,{\displaystyle C^{\textsf {T}}{\bar {C}}\subseteq \ni {\bar {C}}\equiv C{\overline {\ni {\bar {C}}}}\subseteq C,} where∋{\displaystyle \ni } is the converse of set membership (∈{\displaystyle \in }).[51]: 280 

Preorder R\R

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Every relationR{\displaystyle R} generates apreorderR∖R{\displaystyle R\backslash R} which is theleft residual.[52] In terms of converse and complements,R∖R≡RTR¯¯.{\displaystyle R\backslash R\equiv {\overline {R^{\textsf {T}}{\bar {R}}}}.} Forming the diagonal ofRTR¯{\displaystyle R^{\textsf {T}}{\bar {R}}}, the corresponding row ofRT{\displaystyle R^{\textsf {T}}} and column ofR¯{\displaystyle {\bar {R}}} will be of opposite logical values, so the diagonal is all zeros. Then

RTR¯⊆I¯⟹I⊆RTR¯¯=R∖R{\displaystyle R^{\textsf {T}}{\bar {R}}\subseteq {\bar {I}}\implies I\subseteq {\overline {R^{\textsf {T}}{\bar {R}}}}=R\backslash R}, so thatR∖R{\displaystyle R\backslash R} is areflexive relation.

To showtransitivity, one requires that(R∖R)(R∖R)⊆R∖R.{\displaystyle (R\backslash R)(R\backslash R)\subseteq R\backslash R.} Recall thatX=R∖R{\displaystyle X=R\backslash R} is the largest relation such thatRX⊆R.{\displaystyle RX\subseteq R.} Then

R(R∖R)⊆R{\displaystyle R(R\backslash R)\subseteq R}
R(R∖R)(R∖R)⊆R{\displaystyle R(R\backslash R)(R\backslash R)\subseteq R} (repeat)
≡RTR¯⊆(R∖R)(R∖R)¯{\displaystyle \equiv R^{\textsf {T}}{\bar {R}}\subseteq {\overline {(R\backslash R)(R\backslash R)}}} (Schröder's rule)
≡(R∖R)(R∖R)⊆RTR¯¯{\displaystyle \equiv (R\backslash R)(R\backslash R)\subseteq {\overline {R^{\textsf {T}}{\bar {R}}}}} (complementation)
≡(R∖R)(R∖R)⊆R∖R.{\displaystyle \equiv (R\backslash R)(R\backslash R)\subseteq R\backslash R.} (definition)

Theinclusion relation Ω on thepower set ofU{\displaystyle U} can be obtained in this way from themembership relation∈{\displaystyle \in } on subsets ofU{\displaystyle U}:

Ω=∋∈¯¯=∈∖∈.{\displaystyle \Omega ={\overline {\ni {\bar {\in }}}}=\in \backslash \in .}[51]: 283 

Fringe of a relation

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Given a relationR{\displaystyle R}, itsfringe is the sub-relation defined asfringe⁡(R)=R∩RR¯TR¯.{\displaystyle \operatorname {fringe} (R)=R\cap {\overline {R{\bar {R}}^{\textsf {T}}R}}.}

WhenR{\displaystyle R} is a partial identity relation, difunctional, or a block diagonal relation, thenfringe⁡(R)=R{\displaystyle \operatorname {fringe} (R)=R}. Otherwise thefringe{\displaystyle \operatorname {fringe} } operator selects a boundary sub-relation described in terms of its logical matrix:fringe⁡(R){\displaystyle \operatorname {fringe} (R)} is the side diagonal ifR{\displaystyle R} is an upper right triangularlinear order orstrict order.fringe⁡(R){\displaystyle \operatorname {fringe} (R)} is the block fringe ifR{\displaystyle R} is irreflexive (R⊆I¯{\displaystyle R\subseteq {\bar {I}}}) or upper right block triangular.fringe⁡(R){\displaystyle \operatorname {fringe} (R)} is a sequence of boundary rectangles whenR{\displaystyle R} is of Ferrers type.

On the other hand,fringe⁡(R)=∅{\displaystyle \operatorname {fringe} (R)=\emptyset } whenR{\displaystyle R} is adense, linear, strict order.[51]

Mathematical heaps

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Main article:Heap (mathematics)

Given two setsA{\displaystyle A} andB{\displaystyle B}, the set of binary relations between themB(A,B){\displaystyle {\mathcal {B}}(A,B)} can be equipped with aternary operation[a,b,c]=abTc{\displaystyle [a,b,c]=ab^{\textsf {T}}c} wherebT{\displaystyle b^{\mathsf {T}}} denotes theconverse relation ofb{\displaystyle b}. In 1953Viktor Wagner used properties of this ternary operation to definesemiheaps, heaps, and generalized heaps.[53][54] The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:

There is a pleasant symmetry in Wagner's work between heaps, semiheaps, and generalised heaps on the one hand, and groups, semigroups, and generalised groups on the other. Essentially, the various types of semiheaps appear whenever we consider binary relations (and partial one-one mappings) betweendifferent setsA{\displaystyle A} andB{\displaystyle B}, while the various types of semigroups appear in the case whereA=B{\displaystyle A=B}.

— Christopher Hollings, "Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups"[55]

See also

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Notes

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  1. ^Authors who deal with binary relations only as a special case ofn{\displaystyle n}-ary relations for arbitraryn{\displaystyle n} usually writeRxy{\displaystyle Rxy} as a special case ofRx1…xn{\displaystyle Rx_{1}\dots x_{n}} (prefix notation).[8]

References

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  1. ^Meyer, Albert (17 November 2021)."MIT 6.042J Math for Computer Science, Lecture 3T, Slide 2"(PDF).Archived(PDF) from the original on 2021-11-17.
  2. ^abcdefghCodd, Edgar Frank (June 1970)."A Relational Model of Data for Large Shared Data Banks"(PDF).Communications of the ACM.13 (6):377–387.doi:10.1145/362384.362685.S2CID 207549016.Archived(PDF) from the original on 2004-09-08. Retrieved2020-04-29.
  3. ^"Relation definition – Math Insight".mathinsight.org. Retrieved2019-12-11.
  4. ^Enderton 1977, Ch 3. pg. 40
  5. ^Ernst Schröder (1895)Algebra und Logic der Relative, viaInternet Archive
  6. ^C. I. Lewis (1918)A Survey of Symbolic Logic, pages 269–279, via internet Archive
  7. ^Gunther Schmidt, 2010.Relational Mathematics. Cambridge University Press,ISBN 978-0-521-76268-7, Chapt. 5
  8. ^Hans Hermes (1973).Introduction to Mathematical Logic. Hochschultext (Springer-Verlag). London: Springer.ISBN 3540058192.ISSN 1431-4657. Sect.II.§1.1.4
  9. ^Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960].Axiomatic Set Theory. Dover.ISBN 0-486-61630-4.
  10. ^Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in 1996 by Oxford University Press, New York].Set Theory and the Continuum Problem. Dover.ISBN 978-0-486-47484-7.
  11. ^Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover.ISBN 0-486-42079-5.
  12. ^Schmidt, Gunther; Ströhlein, Thomas (2012).Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. Definition 4.1.1.ISBN 978-3-642-77968-8.
  13. ^Christodoulos A. Floudas; Panos M. Pardalos (2008).Encyclopedia of Optimization (2nd ed.). Springer Science & Business Media. pp. 299–300.ISBN 978-0-387-74758-3.
  14. ^abcMichael Winter (2007).Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi.ISBN 978-1-4020-6164-6.
  15. ^G. Schmidt, Claudia Haltensperger, and Michael Winter (1997) "Heterogeneous relation algebra", chapter 3 (pages 37 to 53) inRelational Methods in Computer Science, Advances in Computer Science,Springer booksISBN 3-211-82971-7
  16. ^Jacobson, Nathan (2009),Basic Algebra II (2nd ed.) § 2.1.
  17. ^Garrett Birkhoff & Thomas Bartee (1970)Modern Applied Algebra, page 35, McGraw-Hill
  18. ^Mary P. Dolciani (1962)Modern Algebra: Structure and Method, Book 2, page 339, Houghton Mifflin
  19. ^John C. Baez (6 Nov 2001)."quantum mechanics over a commutative rig".Newsgroup: sci.physics.research.Usenet: 9s87n0$iv5@gap.cco.caltech.edu. RetrievedNovember 25, 2018.
  20. ^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
  21. ^Relative simultaneity at Wikibooks
  22. ^Beth, Thomas;Jungnickel, Dieter;Lenz, Hanfried (1986).Design Theory.Cambridge University Press. p. 15.. 2nd ed. (1999)ISBN 978-0-521-44432-3
  23. ^abcdVan Gasteren 1990, p. 45.
  24. ^abcdeKilp, Knauer, Mikhalev 2000, p. 3.
  25. ^"Functional relation - Encyclopedia of Mathematics".encyclopediaofmath.org. Retrieved2024-06-13.
  26. ^"functional relation in nLab".ncatlab.org. Retrieved2024-06-13.
  27. ^Schmidt 2010, p. 49.
  28. ^Kilp, Knauer, Mikhalev 2000, p. 4.
  29. ^Yao, Y.Y.; Wong, S.K.M. (1995)."Generalization of rough sets using relationships between attribute values"(PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences:30–33..
  30. ^Kunen, Kenneth (1980).Set theory: an introduction to independence proofs. North-Holland. p. 102.ISBN 0-444-85401-0.Zbl 0443.03021.
  31. ^Tarski, Alfred; Givant, Steven (1987).A formalization of set theory without variables. American Mathematical Society. p. 3.ISBN 0-8218-1041-3.
  32. ^M. E. Müller (2012).Relational Knowledge Discovery. Cambridge University Press. p. 22.ISBN 978-0-521-19021-3.
  33. ^Peter J. Pahl; Rudolf Damrath (2001).Mathematical Foundations of Computational Engineering: A Handbook. Springer Science & Business Media. p. 496.ISBN 978-3-540-67995-0.
  34. ^Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),A Transition to Advanced Mathematics (6th ed.), Brooks/Cole, p. 160,ISBN 0-534-39900-2
  35. ^Nievergelt, Yves (2002),Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Springer-Verlag, p. 158.
  36. ^Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007).Transitive Closures of Binary Relations I(PDF). Prague: School of Mathematics – Physics Charles University. p. 1. Archived fromthe original(PDF) on 2013-11-02. Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
  37. ^Joseph G. Rosenstein,Linear orderings, Academic Press, 1982,ISBN 0-12-597680-1, p. 4
  38. ^R. Berghammer & M. Winter (2013) "Decomposition of relations on concept lattices",Fundamenta Informaticae 126(1): 37–82doi:10.3233/FI-2013-871
  39. ^Ki-Hang Kim (1982)Boolean Matrix Theory and Applications, page 37,Marcel DekkerISBN 0-8247-1788-0
  40. ^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 inRelations and Kleene algebras in computer science,Lecture Notes in Computer Science 5827, SpringerMR 2781235
  41. ^Riguet, Jacques (January 1950)."Quelques proprietes des relations difonctionelles".Comptes rendus (in French).230:1999–2000.
  42. ^Julius Richard Büchi (1989).Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions. Springer Science & Business Media. pp. 35–37.ISBN 978-1-4613-8853-1.
  43. ^East, James; Vernitski, Alexei (February 2018). "Ranks of ideals in inverse semigroups of difunctional binary relations".Semigroup Forum.96 (1):21–30.arXiv:1612.04935.doi:10.1007/s00233-017-9846-9.S2CID 54527913.
  44. ^Chris Brink; Wolfram Kahl; Gunther Schmidt (1997).Relational Methods in Computer Science. Springer Science & Business Media. p. 200.ISBN 978-3-211-82971-4.
  45. ^Ali Jaoua, Nadin Belkhiter, Habib Ounalli, and Theodore Moukam (1997) "Databases", pages 197–210 inRelational Methods in Computer Science, edited by Chris Brink, Wolfram Kahl, andGunther Schmidt,Springer Science & Business MediaISBN 978-3-211-82971-4
  46. ^Gumm, H. P.; Zarrad, M. (2014). "Coalgebraic Simulations and Congruences".Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science. Vol. 8446. p. 118.doi:10.1007/978-3-662-44124-4_7.ISBN 978-3-662-44123-7.
  47. ^J. Riguet (1951) "Les relations de Ferrers",Comptes Rendus 232: 1729,30
  48. ^Schmidt, Gunther; Ströhlein, Thomas (2012).Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. p. 77.ISBN 978-3-642-77968-8.
  49. ^Georg Aumann (1971)."Kontakt-Relationen".Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften München.1970 (II):67–77.
  50. ^Anne K. Steiner (1970)Review:Kontakt-Relationen fromMathematical Reviews
  51. ^abcGunther Schmidt (2011)Relational Mathematics, pages 211−15,Cambridge University PressISBN 978-0-521-76268-7
  52. ^In this context, the symbol∖{\displaystyle \backslash } does not mean "set difference".
  53. ^Viktor Wagner (1953) "The theory of generalised heaps and generalised groups",Matematicheskii Sbornik 32(74): 545 to 632MR 0059267
  54. ^C.D. Hollings & M.V. Lawson (2017)Wagner's Theory of Generalised Heaps,Springer booksISBN 978-3-319-63620-7MR 3729305
  55. ^Christopher Hollings (2014)Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups, page 265, History of Mathematics 41,American Mathematical SocietyISBN 978-1-4704-1493-1

Bibliography

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