# 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.