\begin{definition}[Definition:Iterated Binary Operation over Set with Finite Support]
Let $\struct {G, *}$ be a commutative monoid.
Let $S$ be a set.
Let $f: S \to G$ be a mapping.
Let the support $\map \supp f$ be finite.
The '''iteration of $*$ of $f$ over $S$''', denoted $\ds \prod_{s \mathop \in S} \map f s$, is the iteration over the finite set $\map \supp f$ of $f$:
:$\ds \prod_{s \mathop \in S} \map f s = \prod_{s \mathop \in \map \supp f} \map f s$
\end{definition}