Codensity, profiniteness and algebras of semiring-valued measures

Reggio, Luca (2020). Codensity, profiniteness and algebras of semiring-valued measures. Journal of pure and applied algebra, 224(1), pp. 181-205. Elsevier 10.1016/j.jpaa.2019.05.002

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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 Sierpiński space.
We adopt a categorical approach to profinite algebra which is based on profinite monads. The latter were first introduced by Adámek et al. as a special case of the notion of codensity monads.

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