# Definition:Summation/Finite Set

## Definition

### Standard Number System

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.

Let $S$ be a finite set.

Let $f: S \to \mathbb A$ be a mapping.

Let $n$ 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 summation of $f$ over $S$, denoted $\displaystyle \sum_{s \mathop\in S} f(s)$, is the indexed summation of the composition $f \circ g$ over $\N_{<n}$:

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

### Abelian Semigroup

Let $(G, +)$ be an abelian semigroup.

Let:

$S$ be a non-empty finite set

or

$S$ by any finite set and $G$ a abelian monoid.

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

The summation of $f$ over $S$, denoted $\displaystyle \sum_{s \mathop\in S} f(s)$, is an alternative name for the iterated operation of $+$ of $f$ over $S$.

## Remark

Note that by Empty Set is Finite, this definition includes in particular the case where $S$ is empty. See also Summation over Empty Set is Zero.