Codensity, profiniteness and algebras of semiring-valued measures
Publisher DOI
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 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.
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.
Date Issued
2020-01
Publication Type
Article
Subject(s)
Language(s)
en
Author(s)
Additional Credits
Journal
Journal of pure and applied algebra
Publisher
Elsevier
ISSN
0022-4049
Access(Rights)
open.access