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 $S$:
$\odot: S^n \to S: \forall \tuple {s_1, s_2, \ldots, s_n} \in S^n: \map \odot {s_1, s_2, \ldots, s_n} \in S$


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 $S$.


Also see

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

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


Hence:

$\forall \tuple {s_1, s_2, \ldots, s_n} \in S^n: \map \odot {s_1, s_2, \ldots, s_n} := \paren {\cdots \paren {\paren {s_1 \odot s_2} \odot s_3} \odot \cdots} \odot s_n$



Sources