Stone Pseudovarieties

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F24%3A00139702" target="_blank" >RIV/00216224:14310/24:00139702 - isvavai.cz</a>

Result on the web

<a href="https://link.springer.com/article/10.1007/s00025-024-02275-4" target="_blank" >https://link.springer.com/article/10.1007/s00025-024-02275-4</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00025-024-02275-4" target="_blank" >10.1007/s00025-024-02275-4</a>

Result language

angličtina

Original language name

Stone Pseudovarieties

Original language description

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman's theorem can be derived from it.

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10100 - Mathematics

Project

<a href="/en/project/GA19-00902S" target="_blank" >GA19-00902S: Injectivity and Monads in Algebra and Topology</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2024

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Results in Mathematics

ISSN

1422-6383

e-ISSN

1420-9012

Volume of the periodical

79

Issue of the periodical within the volume

7

Country of publishing house

CH - SWITZERLAND

Number of pages

43

Pages from-to

1-43

UT code for WoS article

001319115600001

EID of the result in the Scopus database

2-s2.0-85204909006