# Definition:Operation/Operation on Set

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## Definition

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

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 $S$.

## Also see

$\forall \left({s_1, s_2, \ldots, s_n}\right) \in S^n: \odot \left({s_1, s_2, \ldots, s_n}\right) = \begin{cases} s_1 & : n = 1 \\ \odot \left({s_1, s_2, \ldots, s_{n - 1} }\right) \odot s_n & : n > 1 \end{cases}$

where $\odot \left({s_1, s_2, \ldots, s_{n - 1} }\right)$ is the $n - 1$-ary operation defined in the same way.

Hence:

$\forall \left({s_1, s_2, \ldots, s_n}\right) \in S^n: \odot \left({s_1, s_2, \ldots, s_n}\right) := \left({\cdots\left({\left({s_1 \odot s_2}\right) \odot s_3}\right) \odot \cdots}\right) \odot s_n$