# Homogeneous relation

<https://en.wikipedia.org/wiki/Homogeneous_relation>

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 X × X.\[1]\[2]\[3] 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.

A homogeneous relation R over a set X may be identified with a directed simple graph permitting loops, or if it is symmetric, with an undirected 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). It is called the adjacency relation of the graph.

The set of all homogeneous relations � ( � ) {\displaystyle {\mathcal {B}}(X)} over a set X is the set 2X × X which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on � ( � ) {\displaystyle {\mathcal {B}}(X)}, it forms a semigroup with involution.
