About the power pseudovariety PCS

J 2021

About the power pseudovariety PCS

KAĎOUREK, Jiří

Basic information

Original name

About the power pseudovariety PCS

Authors

KAĎOUREK, Jiří (203 Czech Republic, guarantor, belonging to the institution)

Edition

Rocky Mountain Journal of Mathematics, Rocky Mountain Mathematics Consortium, 2021, 0035-7596

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 0.813

RIV identification code

RIV/00216224:14310/21:00124463

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1216/rmj.2021.51.2045

UT WoS

000772456800012

Keywords in English

aggregates of block groups; block groups; completely simple semigroups; Mal’cev products of pseudovarieties of semigroups; power pseudovarieties; power semigroups of finite semigroups; pseudovarieties of finite semigroups; relatively free semigroups

Abstract

V originále

The power pseudovariety PCS, that is, the pseudovariety of finite semigroups generated by all power semigroups of finite completely simple semigroups has recently been characterized as the pseudovariety AgBG of all so-called aggregates of block groups. This characterization can be expressed as the equality of pseudovarieties PCS=AgBG. In fact, a longer sequence of equalities of pseudovarieties, namely the sequence of equalities PCS=J∗CS=JⓜCS=AgBG has been verified at the same time. Here, J is the pseudovariety of all 𝒥-trivial semigroups, CS is the pseudovariety of all completely simple semigroups, J∗CS is the pseudovariety generated by the family of all semidirect products of 𝒥-trivial semigroups by completely simple semigroups, and JⓜCS is the pseudovariety generated by the Mal’cev product of the pseudovarieties J and CS. In this paper, another different proof of these equalities is provided first. More precisely, the equalities PCS=J∗CS=JⓜCS are given a new proof, while the equality JⓜCS=AgBG is quoted from a foregoing paper. Subsequently in this paper, this new proof of the mentioned equalities is further refined to yield a proof of the following more general result: For any pseudovariety H of groups, let CS(H) stand for the pseudovariety of all completely simple semigroups whose subgroups belong to H. Then it turns out that, for every locally extensible pseudovariety H of groups, the equalities of pseudovarieties P(CS(H))=J∗CS(H)=JⓜCS(H) hold.

Citovat

KAĎOUREK, Jiří. About the power pseudovariety PCS. Rocky Mountain Journal of Mathematics. Rocky Mountain Mathematics Consortium, 2021, vol. 51, No 6, p. 2045-2102. ISSN 0035-7596. Available from: https://dx.doi.org/10.1216/rmj.2021.51.2045.

@article{1843057,
   author = {Kaďourek, Jiří},
   article_number = {6},
   doi = {http://dx.doi.org/10.1216/rmj.2021.51.2045},
   keywords = {aggregates of block groups; block groups; completely simple semigroups; Mal’cev products of pseudovarieties of semigroups; power pseudovarieties; power semigroups of finite semigroups; pseudovarieties of finite semigroups; relatively free semigroups},
   language = {eng},
   issn = {0035-7596},
   journal = {Rocky Mountain Journal of Mathematics},
   title = {About the power pseudovariety PCS},
   url = {https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-51/issue-6/About-the-power-pseudovariety-PCS/10.1216/rmj.2021.51.2045.short},
   volume = {51},
   year = {2021}
}

TY  - JOUR
ID  - 1843057
AU  - Kaďourek, Jiří
PY  - 2021
TI  - About the power pseudovariety PCS
JF  - Rocky Mountain Journal of Mathematics
VL  - 51
IS  - 6
SP  - 2045-2102
EP  - 2045-2102
PB  - Rocky Mountain Mathematics Consortium
SN  - 00357596
KW  - aggregates of block groups
KW  - block groups
KW  - completely simple semigroups
KW  - Mal’cev products of pseudovarieties of semigroups
KW  - power pseudovarieties
KW  - power semigroups of finite semigroups
KW  - pseudovarieties of finite semigroups
KW  - relatively free semigroups
UR  - https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-51/issue-6/About-the-power-pseudovariety-PCS/10.1216/rmj.2021.51.2045.short
N2  - The power pseudovariety PCS, that is, the pseudovariety of finite semigroups generated by all power semigroups of finite completely simple semigroups has recently been characterized as the pseudovariety AgBG of all so-called aggregates of block groups. This characterization can be expressed as the equality of pseudovarieties PCS=AgBG. In fact, a longer sequence of equalities of pseudovarieties, namely the sequence of equalities PCS=J∗CS=JⓜCS=AgBG has been verified at the same time. Here, J is the pseudovariety of all 𝒥-trivial semigroups, CS is the pseudovariety of all completely simple semigroups, J∗CS is the pseudovariety generated by the family of all semidirect products of 𝒥-trivial semigroups by completely simple semigroups, and JⓜCS is the pseudovariety generated by the Mal’cev product of the pseudovarieties J and CS. In this paper, another different proof of these equalities is provided first. More precisely, the equalities PCS=J∗CS=JⓜCS are given a new proof, while the equality JⓜCS=AgBG is quoted from a foregoing paper. Subsequently in this paper, this new proof of the mentioned equalities is further refined to yield a proof of the following more general result: For any pseudovariety H of groups, let CS(H) stand for the pseudovariety of all completely simple semigroups whose subgroups belong to H. Then it turns out that, for every locally extensible pseudovariety H of groups, the equalities of pseudovarieties P(CS(H))=J∗CS(H)=JⓜCS(H) hold.
ER  -

KAĎOUREK, Jiří. About the power pseudovariety PCS. \textit{Rocky Mountain Journal of Mathematics}. Rocky Mountain Mathematics Consortium, 2021, vol.~51, No~6, p.~2045-2102. ISSN~0035-7596. Available from: https://dx.doi.org/10.1216/rmj.2021.51.2045.

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