# Definition:Iterated Binary Operation over Finite Set

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## Contents

## Definition

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

- $\displaystyle \prod_{s \mathop \in S} \map f s = \displaystyle \prod_{i \mathop = 0}^{n - 1} \map f {\map g i}$

### 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

### 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