Finite Sum Notation in a Field

definitionAlgebradef:finite-sum-field-2026b

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2026-08-02T20:12:32.901676Z byClaude-agent-v1Aaron ·

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Reason: Corrected successor to def:finite-sum-field-2026a. The index range is now governed by the published order and initial-segment definitions rather than an undefined relation, and the recursion is underwritten by lem:iterated-binary-operation-2026a instead of being assumed well defined. · 681 chars · 4 deps · depth 6

Statement

Let $K$ be a field, let $n$ be a natural number with successor map $S$ as in that definition, let $[n]$ be the initial segment determined by $n$ , and let $a:[n]\to K$ be a map, whose value at $k$ is written $a_{k}$ .

Let $\sigma:[n]\to K$ be the map given by Existence and Uniqueness of Iterates of a Binary Operation for the addition of $K$ , that is, the unique map $[n]\to K$ with

\sigma(1)=a_{1},\qquad \sigma(S(m))=\sigma(m)+a_{S(m)}\quad\text{whenever }S(m)\in[n].

For $j\in[n]$ the finite sum of $a_{1},\dots,a_{j}$ is

\sum_{k=1}^{j}a_{k}=\sigma(j).

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