Complete semirings

Abstract

In many common semirings there is a possibility of infinite summation, and so we need to consider that possibility too. In particular, a semiring R is complete if and only if to each function f: Ω → R, where Ω is a nonempty set, we can assign a value Σf in R such that the following conditions are satisfied:

• (1) If Ω = {i ₁,..., i _n } is a finite set then Σf = f(i ₁) + ... + f(i _n );

• (2) If r ∈ R then Σ(rf)=r[Σf] and Σ(fr) = [Σf]r;

• (3) If Ω = U _j∈Λ Ω _j is a partition of Ω into a disjoint union of nonempty subsets and if f _j is the restriction of f to Ω _j for each j ∈ Ω then the function g: Λ → R defined by g : j ↦ Σf _j satisfies Σg = Σf. Every complete semiring has an infinite element (see Proposition 22.27 of [215]). For a comprehensive study of infinite sums in semirings, refer to [251].