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

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