Definition:Summation over Finite Index

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

Let $\family{g_i}_{i \mathop \in I}$ 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 $\tuple{g_{e_1}, g_{e_2}, \ldots, g_{e_n}}$ be the ordered tuple formed from the composite mapping $g \circ e: \closedint 1 n \to G$.

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

 * Summation over Finite Index is Well-Defined