J. Almeida, O. Klíma
Author Affiliations +
Publ. Mat. 67(1): 127-172 (2023). DOI: 10.5565/PUBLMAT6712303


The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety V (of finite semigroups) for which all pro-V semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressible in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular J-classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality 1xn is locally countable if and only if n=1.


The authors wish to thank the referees for their thoughtful comments that helped to improve the paper.

The first author acknowledges partial funding by CMUP (UID/MAT/00144/2019), which is funded by FCT (Portugal) with national (MATT’S) and European structural funds (FEDER) under the partnership agreement PT2020. The work was carried out at Masaryk University, whose hospitality is gratefully acknowledged, with the support of the FCT sabbatical scholarship SFRH/BSAB/142872/2018. The second author was supported by grant 19-12790S of the Grant Agency of the Czech Republic.


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J. Almeida. O. Klíma. "LOCALLY COUNTABLE PSEUDOVARIETIES." Publ. Mat. 67 (1) 127 - 172, 2023. https://doi.org/10.5565/PUBLMAT6712303


Received: 31 August 2020; Accepted: 26 January 2021; Published: 2023
First available in Project Euclid: 19 December 2022

MathSciNet: MR4522932
zbMATH: 1515.20289
Digital Object Identifier: 10.5565/PUBLMAT6712303

Primary: 03F03 , 20M05 , 20M07

Keywords: factorization forest , locally countable , Mal’cev product , profinite algebra , Prouhet–Thue–Morse substitution , pseudovariety

Rights: Copyright © 2023 Universitat Autònoma de Barcelona, Departament de Matemàtiques


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Vol.67 • No. 1 • 2023
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