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

[img] Text
Reggio2020.pdf - Accepted Version
Restricted to registered users only until 8 May 2021.
Available under License Publisher holds Copyright.

Download (334kB) | Request a copy
[img] Text
1-s2.0-S0022404919301203-main.pdf - Published Version
Restricted to registered users only
Available under License Publisher holds Copyright.

Download (582kB) | Request a copy

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.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Reggio, Luca

Subjects:

500 Science > 510 Mathematics

ISSN:

0022-4049

Publisher:

Elsevier

Language:

English

Submitter:

George Metcalfe

Date Deposited:

30 Mar 2020 10:04

Last Modified:

30 Mar 2020 10:12

Publisher DOI:

10.1016/j.jpaa.2019.05.002

ArXiv ID:

1807.10637v2

BORIS DOI:

10.7892/boris.141525

URI:

https://boris.unibe.ch/id/eprint/141525

Actions (login required)

Edit item Edit item
Provide Feedback