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All definitions tacitly require the homogeneous relation be transitive: A "✓" indicates that the column property is required in the row definition. For example, the definition of an equivalence relation requires it to be symmetric. Listed here are additional properties that a homogeneous relation may satisfy. |

In mathematics, a **binary relation** over sets X and Y is a subset of the Cartesian product ; that is, it is a set of ordered pairs (*x*, *y*) consisting of elements x in X and y in Y.^{ [1] } It encodes the common concept of relation: an element x is *related* to an element y, if and only if the pair (*x*, *y*) belongs to the set of ordered pairs that defines the *binary relation*. A binary relation is the most studied special case *n* = 2 of an n-ary relation over sets *X*_{1}, ..., *X*_{n}, which is a subset of the Cartesian product ^{ [1] }^{ [2] }

- Definition
- Examples
- Special types of binary relations
- Operations on binary relations
- Union
- Intersection
- Composition
- Converse
- Complement
- Restriction
- Matrix representation
- Sets versus classes
- Homogeneous relation
- Heterogeneous relation
- Calculus of relations
- Induced concept lattice
- Particular relations
- Difunctional
- Ferrers type
- Contact
- Preorder R\R
- Fringe of a relation
- Mathematical heaps
- See also
- Notes
- References
- Bibliography
- External links

An example of a binary relation is the "divides" relation over the set of prime numbers and the set of integers , in which each prime p is related to each integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13.

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

- the "is greater than", "is equal to", and "divides" relations in arithmetic;
- the "is congruent to" relation in geometry;
- the "is adjacent to" relation in graph theory;
- the "is orthogonal to" relation in linear algebra.

A function may be defined as a special kind of binary relation.^{ [3] } Binary relations are also heavily used in computer science.

A binary relation over sets X and Y is an element of the power set of Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of A binary relation is either a homogeneous relation or a heterogeneous relation depending on whether *X* = *Y* or not.

Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the 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,^{ [4] } Clarence Lewis,^{ [5] } and Gunther Schmidt.^{ [6] } 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 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.

The terms *correspondence*,^{ [7] }**dyadic relation** and **two-place relation** are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product without reference to X and Y, and reserve the term "correspondence" for a binary relation with reference to X and Y.^{[ citation needed ]}

Given sets *X* and *Y*, the Cartesian product is defined as and its elements are called ordered pairs.

A *binary relation**R* over sets *X* and *Y* is a subset of ^{ [1] }^{ [8] } The set *X* is called the *domain*^{ [1] } or *set of departure* of *R*, and the set *Y* the *codomain* or *set of destination* of *R*. In order to specify the choices of the sets *X* and *Y*, some authors define a *binary relation* or *correspondence* as an ordered triple (*X*, *Y*, *G*), where *G* is a subset of called the *graph* of the binary relation. The statement reads "*x* is *R*-related to *y*" and is denoted by *xRy*.^{ [4] }^{ [5] }^{ [6] }^{ [note 1] } The *domain of definition* or *active domain*^{ [1] } of *R* is the set of all *x* such that *xRy* for at least one *y*. The *codomain of definition*, *active codomain*,^{ [1] }*image* or *range* of *R* is the set of all *y* such that *xRy* for at least one *x*. The *field* of *R* is the union of its domain of definition and its codomain of definition.^{ [10] }^{ [11] }^{ [12] }

When a binary relation is called a * homogeneous relation * (or *endorelation*). Otherwise it is a * heterogeneous relation *.^{ [13] }^{ [14] }^{ [15] }

In a binary relation, the order of the elements is important; if then *yRx* can be true or false independently of *xRy*. For example, 3 divides 9, but 9 does not divide 3.

AB′ | ball | car | doll | cup |
---|---|---|---|---|

John | + | − | − | − |

Mary | − | − | + | − |

Venus | − | + | − | − |

AB | ball | car | doll | cup |
---|---|---|---|---|

John | + | − | − | − |

Mary | − | − | + | − |

Ian | − | − | − | − |

Venus | − | + | − | − |

1) The following example shows that the choice of codomain is important. Suppose there are four objects and four people A possible relation on *A* and *B* is the relation "is owned by", given by That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing, see 1st example. As a set, *R* does not involve Ian, and therefore *R* could have been viewed as a subset of i.e. a relation over *A* and see 2nd example. While the 2nd example relation is surjective (see below), the 1st is not.

NA | SA | AF | EU | AS | AU | AA | |
---|---|---|---|---|---|---|---|

Indian | 0 | 0 | 1 | 0 | 1 | 1 | 1 |

Arctic | 1 | 0 | 0 | 1 | 1 | 0 | 0 |

Atlantic | 1 | 1 | 1 | 1 | 0 | 0 | 1 |

Pacific | 1 | 1 | 0 | 0 | 1 | 1 | 1 |

2) Let *A* = {Indian, Arctic, Atlantic, Pacific}, the oceans of the globe, and *B* = { NA, SA, AF, EU, AS, AU, AA }, the continents. Let *aRb* represent that ocean *a* borders continent *b*. Then the logical matrix for this relation is:

The connectivity of the planet Earth can be viewed through *R R*^{T} and *R*^{T}*R*, the former being a relation on *A*, which is the universal relation ( or a logical matrix of all ones). This universal relation reflects the fact that every ocean is separated from the others by at most one continent. On the other hand, *R*^{T}*R* is a relation on 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 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 is integral to relations on a set, so bicliques are used to describe heterogeneous relations; indeed, they are the "concepts" that generate a lattice associated with a relation.

4) Hyperbolic orthogonality: Time and space are different categories, and temporal properties are separate from spatial properties. The idea of *simultaneous events* is simple in absolute time and space since each time *t* determines a simultaneous hyperplane in that cosmology. Herman Minkowski changed that when he articulated the notion of *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

- where the overbar denotes conjugation.

As a relation between some temporal events and some spatial events, hyperbolic orthogonality (as found in split-complex numbers) is a heterogeneous relation.^{ [16] }

5) A geometric configuration can be considered a relation between its points and its lines. The relation is expressed as incidence. Finite and infinite projective and affine planes are included. Jakob Steiner pioneered the cataloguing of configurations with the Steiner systems which have an n-element set *S* and a set of k-element subsets called **blocks**, such that a subset with *t* elements lies in just one block. These incidence structures have been generalized with block designs. The incidence matrix used in these geometrical contexts corresponds to the logical matrix used generally with binary relations.

- An incidence structure is a triple
**D**= (*V*,**B**,*I*) where*V*and**B**are any two disjoint sets and*I*is a binary relation between*V*and**B**, i.e. The elements of*V*will be called*points*, those of**B**blocks and those of*I flags*.^{ [17] }

Some important types of binary relations *R* over sets *X* and *Y* are listed below.

*Set-like*^{[ citation needed ]}(or*local*)- for all the class of all y such that
*yRx*is a set. (This makes sense only if relations over proper classes are allowed.) For example, the usual ordering < over the class of ordinal numbers is a set-like relation, while its inverse > is not.

Uniqueness properties:

**Injective**(also called**left-unique**):^{ [18] }for all and all if*xRy*and*zRy*then*x*=*z*. For such a relation, {*Y*} is called*a primary key*of*R*.^{ [1] }For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both −1 and 1 to 1), nor the black one (as it relates both −1 and 1 to 0).**Functional**(also called**right-unique**,^{ [18] }**right-definite**^{ [19] }or**univalent**):^{ [6] }for all and all if*xRy*and*xRz*then*y*=*z*. Such a binary relation is called a*partial function*. For such a relation, is called*a primary key*of*R*.^{ [1] }For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates 1 to both −1 and 1), nor the black one (as it relates 0 to both −1 and 1).**One-to-one**: injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.**One-to-many**: injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.**Many-to-one**: functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.**Many-to-many**: not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not.

Totality properties (only definable if the domain *X* and codomain *Y* are specified):

**Serial**(also called**left-total**):^{ [18] }for all*x*in*X*there exists a*y*in*Y*such that*xRy*. In other words, the domain of definition of*R*is equal to*X*. This property, although also referred to as*total*by some authors,^{[ citation needed ]}is different from the definition of*connected*(also called*total*by some authors)^{[ citation needed ]}in Properties. Such a binary relation is called a*multivalued function*. For example, the red and green binary relations in the diagram are serial, but the blue one is not (as it does not relate −1 to any real number), nor the black one (as it does not relate 2 to any real number). As another example, > is a serial relation over the integers. But it is not a serial relation over the positive integers, because there is no y in the positive integers such that 1 >*y*.^{ [20] }However, < is a serial relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given x, choose*y*=*x*.**Surjective**(also called**right-total**^{ [18] }or**onto**): for all*y*in*Y*, there exists an*x*in*X*such that*xRy*. In other words, the codomain of definition of*R*is equal to*Y*. For example, the green and blue binary relations in the diagram are surjective, but the red one is not (as it does not relate any real number to −1), nor the black one (as it does not relate any real number to 2).

Uniqueness and totality properties (only definable if the domain *X* and codomain *Y* are specified):

- A
*function*: a binary relation that is functional and serial. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not. - An
*injection*: a function that is injective. For example, the green binary relation in the diagram is an injection, but the red, blue and black ones are not. - A
*surjection*: a function that is surjective. For example, the green binary relation in the diagram is a surjection, but the red, blue and black ones are not. - A
*bijection*: a function that is injective and surjective. For example, the green binary relation in the diagram is a bijection, but the red, blue and black ones are not.

If *R* and *S* are binary relations over sets *X* and *Y* then is the *union relation* of *R* and *S* over *X* and *Y*.

The identity element is the empty relation. For example, is the union of < and =, and is the union of > and =.

If *R* and *S* are binary relations over sets *X* and *Y* then is the *intersection relation* of *R* and *S* over *X* and *Y*.

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

If *R* is a binary relation over sets *X* and *Y*, and *S* is a binary relation over sets *Y* and *Z* then (also denoted by *R*; *S*) is the *composition relation* of *R* and *S* over *X* and *Z*.

The identity element is the identity relation. The order of *R* and *S* in the notation used here agrees with the standard notational order for composition of functions. For example, the composition "is mother of" "is parent of" yields "is maternal grandparent of", while the composition "is parent of" "is mother of" yields "is grandmother of". For the former case, if *x* is the parent of *y* and *y* is the mother of *z*, then *x* is the maternal grandparent of *z*.

If *R* is a binary relation over sets *X* and *Y* then is the *converse relation* of *R* over *Y* and *X*.

For example, = is the converse of itself, as is and and are each other's converse, as are and A binary relation is equal to its converse if and only if it is symmetric.

If *R* is a binary relation over sets *X* and *Y* then (also denoted by or ¬*R**R*) is the *complementary relation* of *R* over *X* and *Y*.

For example, and are each other's complement, as are and and and and and, for total orders, also < and and > and

The complement of the converse relation is the converse of the complement:

If 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.

If *R* is a binary homogeneous relation over a set *X* and *S* is a subset of *X* then is the *restriction relation* of *R* to *S* over *X*.

If *R* is a binary relation over sets *X* and *Y* and if *S* is a subset of *X* then is the *left-restriction relation* of *R* to *S* over *X* and *Y*.

If *R* is a binary relation over sets *X* and *Y* and if *S* is a subset of *Y* then is the *right-restriction relation* of *R* to *S* over *X* and *Y*.

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, 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 "*x* is parent of *y*" to females yields the relation "*x* is mother of the woman *y*"; its transitive closure doesn't 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 (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers a property of the relation is that every non-empty subset with an upper bound in has a least upper bound (also called supremum) in However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation to the rational numbers.

A binary relation *R* over sets *X* and *Y* is said to be *contained in* a relation *S* over *X* and *Y*, written if *R* is a subset of *S*, that is, for all and if *xRy*, then *xSy*. If *R* is contained in *S* and *S* is contained in *R*, then *R* and *S* are called *equal* written *R* = *S*. If *R* is contained in *S* but *S* is not contained in *R*, then *R* is said to be *smaller* than *S*, written For example, on the rational numbers, the relation is smaller than and equal to the composition

Binary relations over sets *X* and *Y* can be represented algebraically by logical matrices indexed by *X* and *Y* 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 *X* and *Y* and a relation over *Y* and *Z*),^{ [21] } the 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 *X* = *Y*) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.^{ [22] }

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 take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory.

In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set *A*, that contains all the objects of interest, and work with the restriction =_{A} instead of =. Similarly, the "subset of" relation needs to be restricted to have domain and codomain P(*A*) (the power set of a specific set *A*): the resulting set relation can be denoted by Also, the "member of" relation needs to be restricted to have domain *A* and codomain P(*A*) to obtain a binary relation that is a set. Bertrand Russell has shown that assuming to be defined over all sets leads to a contradiction in naive set theory.

Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory, and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple (*X*, *Y*, *G*), as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.)^{ [23] } With this definition one can for instance define a binary relation over every set and its power set.

A **homogeneous relation** over a set *X* is a binary relation over *X* and itself, i.e. it is a subset of the Cartesian product ^{ [15] }^{ [24] }^{ [25] } It is also simply called a (binary) relation over *X*.

A homogeneous relation *R* over a set *X* may be identified with a directed simple graph permitting loops, where *X* is the vertex set and *R* is the edge set (there is an edge from a vertex *x* to a vertex *y* if and only if *xRy*). The set of all homogeneous relations over a set *X* is the power set 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 semigroup with involution.

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

*Reflexive*: for all*xRx*. For example, is a reflexive relation but > is not.*Irreflexive*: for all not*xRx*. For example, is an irreflexive relation, but is not.*Symmetric*: for all if*xRy*then*yRx*. For example, "is a blood relative of" is a symmetric relation.*Antisymmetric*: for all if*xRy*and*yRx*then For example, is an antisymmetric relation.^{ [26] }*Asymmetric*: for all if*xRy*then not*yRx*. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.^{ [27] }For example, > is an asymmetric relation, but is not.*Transitive*: for all if*xRy*and*yRz*then*xRz*. A transitive relation is irreflexive if and only if it is asymmetric.^{ [28] }For example, "is ancestor of" is a transitive relation, while "is parent of" is not.*Connected*: for all if then*xRy*or*yRx*.*Strongly connected*: for all*xRy*or*yRx*.

A * partial order * is a relation that is reflexive, antisymmetric, and transitive. A * strict partial order * is a relation that is irreflexive, antisymmetric, and transitive. A * total order * is a relation that is reflexive, antisymmetric, transitive and connected.^{ [29] } A * strict total order * is a relation that is irreflexive, antisymmetric, transitive and connected. An * equivalence relation * is a relation that is reflexive, symmetric, and transitive. For example, "*x* divides *y*" is a partial, but not a total order on natural numbers "*x* < *y*" is a strict total order on and "*x* is parallel to *y*" is an equivalence relation on the set of all lines in the Euclidean plane.

All operations defined in the section Operations on binary relations also apply to homogeneous relations. Beyond that, a homogeneous relation over a set *X* may be subjected to closure operations like:

*Reflexive closure*:the (unique) reflexive relation over*X*containing*R*,*Transitive closure*: the smallest transitive relation over*X*containing*R*,*Equivalence closure*: the smallest equivalence relation over*X*containing*R*.

In mathematics, a **heterogeneous relation** is a binary relation, a subset of a Cartesian product where *A* and *B* are distinct sets.^{ [30] } The prefix *hetero* is from the Greek ἕτερος (*heteros*, "other, another, different").

A heterogeneous relation has been called a **rectangular relation**,^{ [15] } suggesting that it does not have the square-symmetry of a homogeneous relation on a set where 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 as *heterogeneous* or *rectangular*, i.e. as relations where the normal case is that they are relations between different sets."^{ [31] }

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 relations. The inclusion meaning that *aRb* implies *aSb*, sets the scene in a lattice of relations. But since the inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of

In contrast to homogeneous relations, the composition of relations operation is only a partial function. The necessity of matching range to domain 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 morphisms of this category are relations. The *objects* of the category Rel are sets, and the relation-morphisms compose as required in a category.^{[ citation needed ]}

Binary relations have been described through their induced concept lattices: A **concept***C* ⊂ *R* satisfies two properties: (1) The logical matrix of *C* is the outer product of logical vectors

- logical vectors.
^{[ clarification needed ]}(2)*C*is maximal, not contained in any other outer product. Thus*C*is described as a non-enlargeable rectangle.

For a given relation the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion 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".^{ [32] } The decomposition is

- where
*f*and*g*are functions, called*mappings*or left-total, univalent relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial order*E*that belongs to the minimal decomposition (*f, g, E*) of the relation*R*."

Particular cases are considered below: *E* total order corresponds to Ferrers type, and *E* identity corresponds to difunctional, a generalization 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.^{ [33] } Structural analysis of relations with concepts provides an approach for data mining.^{ [34] }

*Proposition*: If*R*is a serial relation and R^{T}is its transpose, then where is the*m*×*m*identity relation.*Proposition*: If*R*is a surjective relation, then where is the identity relation.

Among the homogeneous relations on a set, the equivalence relations are distinguished for their ability to partition the set. With binary relations in general the idea is to partition objects by distinguishing attributes. One way this can be done is with an intervening set of indicators. The partitioning relation is a composition of relations using *univalent* relations

The logical matrix of such a relation *R* can be re-arranged as a block matrix with blocks of ones along the diagonal. In terms of the calculus of relations, in 1950 Jacques Riguet showed that such relations satisfy the inclusion

^{ [35] }

He named these relations **difunctional** since the composition *F G*^{T} involves univalent relations, commonly called *functions*.

Using the notation {*y*: *xRy*} = *xR*, a difunctional relation can also be characterized as a relation *R* such that wherever *x*_{1}*R* and *x*_{2}*R* have a non-empty intersection, then these two sets coincide; formally implies ^{ [36] }

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management."^{ [37] } Furthermore, difunctional relations are fundamental in the study of bisimulations.^{ [38] }

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

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 diagonal matrix with rectangular blocks of true on the (asymmetric) main diagonal.^{ [39] }

A strict order on a set is a homogeneous relation arising in order theory. In 1951 Jacques Riguet adopted the ordering of a partition of an integer, called a Ferrers diagram, to extend ordering to binary relations in general.^{ [40] }

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

If any one of the relations is of Ferrers type, then all of them are. ^{ [30] }

Suppose *B* is the power set of *A*, the set of all subsets of *A*. Then a relation *g* is a **contact relation** if it satisfies three properties:

The set membership relation, ε = "is an element of", satisfies these properties so ε is a contact relation. The notion of a general contact relation was introduced by Georg Aumann in 1970.^{ [41] }^{ [42] }

In terms of the calculus of relations, sufficient conditions for a contact relation include

where is the converse of set membership (∈).^{ [43] }^{: 280 }

Every relation *R* generates a preorder which is the left residual.^{ [44] } In terms of converse and complements, Forming the diagonal of , the corresponding row of and column of will be of opposite logical values, so the diagonal is all zeros. Then

- so that is a reflexive relation.

To show transitivity, one requires that Recall that is the largest relation such that Then

- (repeat)
- (Schröder's rule)
- (complementation)
- (definition)

The inclusion relation Ω on the power set of *U* can be obtained in this way from the membership relation on subsets of *U*:

^{ [43] }^{: 283 }

Given a relation *R*, a sub-relation called its *fringe* is defined as

When *R* is a partial identity relation, difunctional, or a block diagonal relation, then fringe(*R*) = *R*. Otherwise the fringe operator selects a boundary sub-relation described in terms of its logical matrix: fringe(*R*) is the side diagonal if *R* is an upper right triangular linear order or strict order. Fringe(*R*) is the block fringe if R is irreflexive () or upper right block triangular. Fringe(*R*) is a sequence of boundary rectangles when *R* is of Ferrers type.

On the other hand, Fringe(*R*) = ∅ when *R* is a dense, linear, strict order.^{ [43] }

Given two sets *A* and *B*, the set of binary relations between them can be equipped with a ternary operation where *b*^{T} denotes the converse relation of *b*. In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps, heaps, and generalized heaps.^{ [45] }^{ [46] } 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) between

differentsetsAandB, while the various types of semigroups appear in the case whereA=B.— Christopher Hollings, "Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups"^{ [47] }

- 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

- ↑ Authors who deal with binary relations only as a special case of
*n*-ary relations for arbitrary*n*usually write*Rxy*as a special case of*Rx*_{1}...*x*_{n}(prefix notation).^{ [9] }

In mathematics, an **equivalence relation** is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation.

In mathematics, a **finitary relation** over sets *X*_{1}, …, *X*_{n} is a subset of the Cartesian product *X*_{1} × … × *X*_{n}; that is, it is a set of *n*-tuples (*x*_{1}, …, *x*_{n}) consisting of elements *x*_{i} in *X*_{i}. Typically, the relation describes a possible connection between the elements of an *n*-tuple. For example, the relation "*x* is divisible by *y* and *z*" consists of the set of 3-tuples such that when substituted to *x*, *y* and *z*, respectively, make the sentence true.

In mathematics, especially order theory, a **partially ordered set** formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word *partial* in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.

In set theory, the **complement** of a set A, often denoted by *A*^{c}, are the elements not in A.

In mathematics, a homogeneous binary relation *R* on a set *X* is **reflexive** if it relates every element of *X* to itself.

In mathematics, a relation *R* on a set *X* is **transitive** if, for all elements *a*, *b*, *c* in *X*, whenever *R* relates *a* to *b* and *b* to *c*, then *R* also relates *a* to *c*. Each partial order as well as each equivalence relation needs to be transitive.

In mathematics, the **transitive closure** of a binary relation *R* on a set *X* is the smallest relation on *X* that contains *R* and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets it is the unique minimal transitive superset of *R*.

In mathematics, a set is **closed** under an operation if performing that operation on members of the set always produces a member of that set. For example, the positive integers are closed under addition, but not under subtraction: 1 − 2 is not a positive integer even though both 1 and 2 are positive integers. Another example is the set containing only zero, which is closed under addition, subtraction and multiplication.

In mathematics, a **principal homogeneous space**, or **torsor**, for a group *G* is a homogeneous space *X* for *G* in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group *G* is a non-empty set *X* on which *G* acts freely and transitively . An analogous definition holds in other categories, where, for example,

This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:

In mathematics, the **image** of a function is the set of all output values it may produce.

In mathematics, the **restriction** of a function is a new function, denoted or , obtained by choosing a smaller domain *A* for the original function .

In mathematics, a **partial equivalence relation** on a set is a binary relation that is *symmetric* and *transitive*. In other words, it holds for all that:

- if , then (symmetry)
- if and , then (transitivity)

In mathematics, the **converse relation**, or **transpose**, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if and are sets and is a relation from to then is the relation defined so that if and only if In set-builder notation,

In the mathematics of binary relations, the **composition of relations** is the forming of a new binary relation *R* ; *S* from two given binary relations *R* and *S*. In the calculus of relations, the composition of relations is called **relative multiplication**, and its result is called a **relative product**. Function composition is the special case of composition of relations where all relations involved are functions.

In set theory, a branch of mathematics, a **serial relation**, also called a **total** or more specifically **left-total relation**, is a binary relation *R* for which every element of the domain has a corresponding range element.

In mathematics, a **homogeneous relation** over a set *X* is a binary relation over *X* and itself, i.e. it is a subset of the Cartesian product *X* × *X*. It is also simply called a (binary) relation over *X*. An example of a homogeneous relation is the relation of kinship, where the relation is over people.

In the mathematical field of category theory, an **allegory** is a category that has some of the structure of the category **Rel** of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation algebra to relations between different sorts. Allegories are also useful in defining and investigating certain constructions in category theory, such as exact completions.

In mathematics, a **binary relation** over sets X and Y is a subset of the Cartesian product *X* × *Y*; that is, it is a set of ordered pairs (*x*, *y*) consisting of elements x in X and y in Y. It encodes the common concept of relation: an element x is *related* to an element y, if and only if the pair (*x*, *y*) belongs to the set of ordered pairs that defines the *binary relation*. A binary relation is the most studied special case *n* = 2 of an n-ary relation over sets *X*_{1}, ..., *X*_{n}, which is a subset of the Cartesian product *X*_{1} × ... × *X*_{n}.

In order theory, the **Szpilrajn extension theorem**, proved by Edward Szpilrajn in 1930, states that every strict partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable. The theorem is one of many examples of the use of the axiom of choice in the form of Zorn's lemma to find a maximal set with certain properties.

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