Definition:Endorelation

Definition
Let $S \times S$ be the cartesian product of a set $S$ 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, or a relation in $S$, or a relation on $S$.

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

Generalized Definition
A subset of a cartesian space $S^n$ is an $n$-ary endorelation on $S$, or just an $n$-ary relation on $S$.

Note
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.