# Definition:Operation/N-Ary Operation

## Definition

Let $S_1, S_2, \dots, S_n$ be sets.

Let $\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb U$ be a mapping from the cartesian product $S_1 \times S_2 \times \ldots \times S_n$ to a universal set $\mathbb U$:

That is, suppose that:

$\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb U: \forall \tuple {s_1, s_2, \ldots, s_n} \in S_1 \times S_2 \times \ldots \times S_n: \map \circ {s_1, s_2, \ldots, s_n} \in \mathbb U$

Then $\circ$ is an $n$-ary operation.

## Remark

An $n$-ary operation needs to be defined for all ordered tuples in $S_1 \times S_2 \times \ldots \times S_n$.

## Also known as

An $n$-ary operation is also sometimes referred to as a finitary operation, although the latter term may also encompass multiary operations as well.