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:


 * $\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb U: \forall \left({s_1, s_2, \ldots, s_n}\right) \in S_1 \times S_2 \times \ldots \times S_n: \circ \left({s_1, s_2, \ldots, s_n}\right) \in \mathbb U$

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 referred to as a finitary operation.

Also see

 * Definition:Operation