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 $\set{i_1, i_2, \ldots, i_n}$ be a finite enumeration of $I$.

Let $\tuple{g_{i_1}, g_{i_2}, \ldots, g_{i_n}}$ be the ordered tuple formed from $\set{i_1, i_2, \ldots, i_n}$.

The summation over $I$, denoted $\ds \sum_{i \mathop \in I} g_i$, is defined as the summation over $\tuple{g_{i_1}, g_{i_2}, \ldots, g_{i_n}}$:
 * $\ds \sum_{i \mathop \in I} g_i = \sum_{k \mathop = 1}^n g_{i_k}$

Also see

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