empty relation

In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation over ''X''". An example of a homogeneous relation is the relation of kinship, where the relation is over people.

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

Particular homogeneous relations

Some particular homogeneous relations over a set ''X'' (with arbitrary elements , ) are:
*Empty relation
*:;
that is, holds never;
*Universal relation
*:;
that is, holds always;
*Identity relation (see also identity function)
*:;
that is, holds if and only if .

Example

Fifteen 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 indicating contact and 0 no contact. This example expresses a symmetric relation.

Properties

Some important properties that a homogeneous relation over a set may have are:
; : for all , . For example, ≥ is a reflexive relation but > is not.
; (or ): for all , not . For example, > is an irreflexive relation, but ≥ is not.
; : for all , if then . 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.
; : for all , if then .
; : for all , if then .
; : for all , if then and . 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 red binary relation is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair , and , but not , respectively. The latter two facts also rule out (any kind of) quasi-reflexivity.

; : for all , if then . For example, "is a blood relative of" is a symmetric relation, because is a blood relative of if and only if is a blood relative of .
; : for all , if and then . For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false).
; : for all , if then not . A relation is asymmetric if and only if it is both antisymmetric and irreflexive. 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 defined by is neither symmetric nor antisymmetric, let alone asymmetric.
; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
; : for all , if and then never .
; : if the complement of ''R'' is transitive. That is, for all , if , then or . This is used in pseudo-orders in constructive mathematics.
; : for all , if and but neither nor , then but not .
; : for all , if and are incomparable with respect to and if the same is true of and , then and are also incomparable with respect to . This is used in weak orderings.
Again, the previous 5 alternatives are not exhaustive. For example, the relation if ( or ) satisfies none of these properties. On the other hand, the empty relation trivially satisfies all of them.

; : for all such that , there exists some such that and . This is used in dense orders.
; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "left/right-total" given below.
; : for all , or . This property, too, is sometimes called "total", which is distinct from the definitions of "left/right-total" given below.
; : for all , exactly one of , or holds. For example, > is a trichotomous relation, while the relation "divides" over the natural numbers is not.
; (or just ): for all , if and then . For example, = is a Euclidean relation because if and then .
; : for all , if and then .
; : every nonempty subset of contains a minimal element with respect to . Well-foundedness implies the descending chain condition (that is, no infinite chain ... can exist). If the axiom of dependent choice is assumed, both conditions are equivalent.

Moreover, all properties of binary relations in general also may apply to homogeneous relations:
; :for all , the class of all such that is a set. (This makes sense only if relations over proper classes are allowed.)
; : for all and all , if and then .
; : for all and all , if and then .
; (also called left-total): for all there exists a such that . This property is different from the definition of ''connected'' (also called ''total'' by some authors).
; (also called right-total): for all , there exists an such that ''xRy''.

A is a relation that is reflexive and transitive. A , also called or , is a relation that is reflexive, transitive, and connected.

A , also called , is a relation that is reflexive, antisymmetric, and transitive. A , also called , is a relation that is irreflexive, antisymmetric, and transitive. A , also called , , or , is a relation that is reflexive, antisymmetric, transitive and connected. A , also called , , or , is a relation that is irreflexive, antisymmetric, transitive and connected.

A is a relation that is symmetric and transitive. An 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.

Operations

If ''R'' is a homogeneous relation over a set ''X'' then each of the following is a homogeneous relation over ''X'':
; , ''R''⁼: Defined as ''R''⁼ = ∪ ''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''.
; , ''R''^≠: Defined as ''R''^≠ = ''R'' \ or the largest irreflexive relation over ''X'' contained in ''R''.
; , ''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''.
; , ''R''*: Defined as , the smallest preorder containing ''R''.
; , ''R''^≡: Defined as the smallest equivalence relation over ''X'' containing ''R''.

All operations defined in also apply to homogeneous relations.

:

Enumeration

The set of all homogeneous relations $\mathcal(X)$ over a set ''X'' is the 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 $\mathcal(X)$, it forms a monoid with involution where the identity element is the identity relation.

The number of distinct homogeneous relations over an ''n''-element set is :

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 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 ''R'' ⊆ ''X'' × ''Y'' for arbitrary sets ''X'' and ''Y''.
* A finitary relation is a subset ''R'' ⊆ ''X''₁ × ... × ''X''_''n'' for some natural number ''n'' and arbitrary sets ''X''₁, ..., ''X''_''n'', it is also called an ''n''-ary relation.

References

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Binary relations