\begin{definition}[Definition:Iterated Binary Operation over Finite Set]
Let $\struct {G, *}$ be a commutative semigroup.
Let $S$ be a finite non-empty set.
Let $f: S \to G$ be a mapping.
Let $n \in \N$ be the cardinality of $S$.
Let $g: \N_{<n} \to S$ be a bijection, where $\N_{<n}$ is an initial segment of the natural numbers.
The '''iteration of $*$ of $f$ over $S$''', denoted $\displaystyle \prod_{s \mathop \in S} \map f s$, is the indexed iteration of $*$ of the composition $f \circ g$ over $\N_{<n}$:
:$\displaystyle \prod_{s \mathop \in S} \map f s = \displaystyle \prod_{i \mathop = 0}^{n - 1} \map f {\map g i}$
\end{definition}