Definition:Iterated Binary Operation over Set with Finite Support

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Let $(G, *)$ be a commutative monoid.

Let $S$ be a set.

Let $f : S \to G$ be a mapping.

Let the support $\operatorname{Supp}f$ be finite.

The iteration of $*$ of $f$ over $S$, denoted $\displaystyle\prod_{s\mathop\in S} f(s)$, is the iteration over the finite set $\operatorname{Supp} f$ of $f$:

$\displaystyle\prod_{s\mathop\in S} f(s) = \displaystyle\prod_{s \mathop\in \operatorname{Supp}f} f(s)$

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Special cases