Definition:Endorelation

Definition
Let $S \times S$ be the cartesian product of a set or class $S$ with itself.

Let $\mathcal R \subseteq S \times S$ be a relation on $S \times S$.

Then $\mathcal R$ is referred to as an endorelation on $S$.

Also known as
The term endorelation is rarely seen. Once it is established that the domain and codomain of a given relation are the same set, further comment is rarely needed.

An endorelation is also called a relation in $S$, or a relation on $S$. The latter term is discouraged, though, because it can also mean a left-total relation, and confusion can arise.

Some sources use the term binary relation exclusively to refer to a binary endorelation.

Some sources, for example, use the term relation for what is defined here as an endorelation, and a relation defined as a general ordered triple of sets: $\left({S, T, \mathcal R \subseteq S \times T}\right)$ is called a correspondence.

As this can cause confusion with the usage of correspondence to mean a relation which is both left-total and right-total, it is recommended that this is not used.