Formal sums

Generally, formal sums are purely formal, i.e. notational, invocations of infinite sums, without considering convergence &c. Specifically, a formal sum fcalc

\sum 𝑠 \in 𝑆 𝑎 𝑠 𝑥 𝑠 \in 𝑅 𝑆; 𝑥 ≅ 𝑅 𝑆

with coëfficients in $𝑅$ and exponents in $𝑆$ may be identified with a function

𝑎 : 𝑆 \to 𝑅 𝑠 \mapsto 𝑎 𝑠

The formal variable $𝑥$ is then simply a bookkeeping device. In most applications, $𝑅$ is a commutative monoid, in which case $𝑅 𝑆; 𝑥$ is a commutative monoid under pointwise addition, and in many cases the structure is richer.

Special cases

Series ring: For a ring $𝑅$ , we have $𝑅 [[𝑥]] = 𝑅 ℕ 0; 𝑥$
Formal sums over a vector space: $𝑉 {𝑥} = 𝑉 𝕂; 𝑥$

Table of Contents

Formal sums

Special cases

See also

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