\begin{definition}[Definition:Operation/Operation on Set]
An '''$n$-ary operation on a set $S$''' is an $n$-ary operation where:
:the domain is the cartesian space $S^n$
:the codomain is $S$:
:$\odot: S^n \to S: \forall \tuple {s_1, s_2, \ldots, s_n} \in S^n: \map \odot {s_1, s_2, \ldots, s_n} \in S$
That is:
:an '''$n$-ary operation on $S$''' needs to be defined for ''all'' tuples in $S^n$
:the image of $\odot$ is itself in $S$.
\end{definition}