Codensity, profiniteness and algebras of semiring-valued measures

TL;DR

This paper characterizes free profinite semimodules over Boolean spaces as algebras of finitely additive S-valued measures, generalizing classical hyperspace representations and employing a categorical approach with profinite monads.

Contribution

It provides a categorical characterization of free profinite S-semimodules as measure algebras, extending classical results to semiring-valued measures and using profinite monads.

Findings

Free profinite S-semimodules are isomorphic to S-valued measure algebras.

S-valued measures are given by continuous density functions when S is (pro)finite.

Generalizes classical hyperspace representations via continuous functions.

Abstract

We show that, if S is a finite semiring, then the free profinite S-semimodule on a Boolean Stone space X is isomorphic to the algebra of all S-valued measures on X, which are finitely additive maps from the Boolean algebra of clopens of X to S. These algebras naturally appear in the logic approach to formal languages as well as in idempotent analysis. Whenever S is a (pro)finite idempotent semiring, the S-valued measures are all given uniquely by continuous density functions. This generalises the classical representation of the Vietoris hyperspace of a Boolean Stone space in terms of continuous functions into the Sierpinski space. We adopt a categorical approach to profinite algebra which is based on profinite monads. The latter were first introduced by Adamek et al. as a special case of the notion of codensity monads.