Completion of Semirings * Definitions and Easy Facts

A semiring can be “completed” (i.e., embedded into a semiring in which all infinite sums are defined and satisfy some reasonable properties) iff this semiring can be naturally partially ordered. This construction is “natural” (a left adjoint to the forgetful functor), and quite straightforward. Definitions and easy facts Definition 1. A semiring is a structure (S,+, ·, 0, 1) such that (S,+, 0) is a commutative monoid, (S, ·, 1) is a monoid, and the two distributive laws x · (y+ z) = x · y+ x · z, (y+ z) ·x = y ·x+ z ·x hold. Definition 2. A complete semiring (S,+, ·, 0, 1,Σ) is a semiring in which for any family (ai : i ∈ I) the infinite sum ∑ i∈I ai is defined, and the function ∑ satisfies the following: • If I = {j, j} has two elements, then ∑ i∈I ai = aj + aj′ • If f : I → J is a bijection, and ai = bf(i) for all i ∈ I, then ∑