Definition:Summation over Finite Index

Definition
Let $\struct{G, +}$ be a commutative monoid.

Let $\family{g }_{i \mathop \in I}$ be an indexed subset of $G$ where the indexing set $I$ is finite.

Let $\set{e_1, e_2, \ldots, e_n}$ be a finite enumeration of $I$.

Let $g: I \to G$ denote the mapping from $I$ to $G$ that defines $\family{g }_{i \mathop \in I}$,

Let $e: \closedint 1 n \to I$ denote the bijection from $\closedint 1 n$ to $I$ that defines $\set{e_1, e_2, \ldots, e_n}$.

Let $g \circ e: \closedint 1 n \to G$ denote the composite of $g$ with $e$.

For each $k \in \closedint 1 n$, let:
 * $g_{e_k} = \map {g \circ e} k$

Let $\tuple{g_{e_1}, g_{e_2}, \ldots, g_{e_n}}$ be the ordered tuple formed from $\set{g_{e_1}, g_{e_2}, \ldots, g_{e_n}}$.

The summation over $I$, denoted $\ds \sum_{i \mathop \in I} g_i$, is defined as the summation over $\tuple{g_{e_1}, g_{e_2}, \ldots, g_{e_n}}$:
 * $\ds \sum_{i \mathop \in I} g_i = \sum_{k \mathop = 1}^n g_{e_k}$

Also see

 * User:Leigh.Samphier/Topology/Summation over Finite Index is Well-Defined