# Definition:Endorelation/General Definition

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## Definition

An $n$-ary relation $\RR$ on a cartesian space $S^n$ is an **$n$-ary endorelation on $S$**:

- $\RR = \struct {S, S, \ldots, S, R}$

where $R \subseteq S^n$.

## 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 **$n$-ary endorelation** is also called an **$n$-ary relation in $S$**, or **on $S$**.

The **on $S$** form is discouraged, though, because it can also mean a left-total relation, and confusion can arise.

## Sources

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations