Definition:Iterated Binary Operation over Finite Set

Definition
Let $\left({G, *}\right)$ 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 $\displaystyle \prod_{s \mathop \in S} f \left({s}\right)$, is the indexed iteration of $*$ of the composition $f \circ g$ over $\N_{< n}$:


 * $\displaystyle \prod_{s \mathop \in S} f \left({s}\right) = \displaystyle \prod_{i \mathop = 0}^{n - 1} f \left({g \left({i}\right)}\right)$

Commutative Monoid
Let $G$ be a commutative monoid.

Let $S$ be a non-empty set.

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

Also known as
The iterated binary operation over a finite set can be referred to as the summation over a finite set.

Also see

 * Iteration of Operation over Finite Set is Well-Defined

Special cases

 * Definition:Indexed Iterated Binary Operation, as shown at Iteration of Operation over Interval equals Indexed Iteration
 * Definition:Summation
 * Definition:Product over Finite Set

Generalizations

 * Definition:Iterated Binary Operation over Set with Finite Support