Definition:Iterated Binary Operation over Set with Finite Support

Definition
Let $\struct {G, *}$ 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 $\ds \prod_{s \mathop \in S} \map f s$, is the iteration over the finite set $\operatorname{Supp} f$ of $f$:
 * $\ds \prod_{s \mathop \in S} \map f s = \prod_{s \mathop \in \operatorname {Supp} f} \map f s$

Special cases

 * Definition:Summation over Set with Finite Support
 * Definition:Product over Set with Finite Support
 * Definition:Iterated Binary Operation over Finite Set, as shown at Iterated Operation over Finite Set equals Iteration over Support