Definition:Operation/Operation on Set

Definition
An $n$-ary operation on a set $S$ is an $n$-ary operation where the domain is the cartesian space $S^n$ and the codomain is $S$:


 * $\odot: S^n \to S: \forall \left({s_1, s_2, \ldots, s_n}\right) \in S^n: \odot \left({s_1, s_2, \ldots, s_n}\right) \in S$

defined as:


 * $\odot \left({s_1, s_2, \ldots, s_k}\right) = \begin{cases}

s_1 & : k = 1 \\ \odot \left({s_1, s_2, \ldots, s_n}\right) \odot s_{n + 1} & : k = n + 1 \end{cases}$

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 $S$.