Congruences on abundant semigroups associated with {G}reen's \(*\)-relations
type: Item
instance of: scholarly article
title: Congruences on abundant semigroups associated with {G}reen's \(*\)-relations
zbMATH Open document ID: 1399.20067
author: Aiqin Liu
author: Xiaojiang Guo
published in: Periodica Mathematica Hungarica
publication date: 2018-02-13T00:00:00Z
review text: \textit{F. Pastijn} and \textit{M. Petrich} [Boll. Unione Mat. Ital., VII. Ser., B 1, 591--603 (1987; Zbl 0621.20036)] comprehensively studied the congruences on regular semigroups associated with Green's relations. The purpose of this paper is to generalize those results to ``good'' congruences on abundant semigroups. Recall that a semigroup \(S\) is \textit{abundant} if each class of the generalized Green relations \(\mathcal L^\ast\) and \(\mathcal R^\ast\) contains an idempotent; the good congruences are those that respect \(\mathcal L^\ast\) and \(\mathcal R^\ast\), a natural restriction in this context. Direct analogues are found for many of the results of Pastijn and Petrich [loc. cit.]. As an example, the least good congruence generated by \(\mathcal L^\ast\) is the least good congruence whose quotient is a right regular band. The cited work also considered congruences on regular semigroups generated by the restrictions of Green's relations to the idempotents. These results are also generalized, although in order to do so the authors have to impose a further condition on an abundant semigroup: that the regular elements form a subsemigroup.
reviewed by: Peter R. Jones
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 6837918
zbMATH Keywords: abundant semigroup
zbMATH Keywords: Green's \(*\)-relations
zbMATH Keywords: good congruence
zbMATH Keywords: congruence lattice
MaRDI profile type: Publication
full work available at URL: s10998-016-0163-y
OpenAlex ID: W2549922372
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DOI: 10.1007/S10998-016-0163-Y
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description: scientific article; zbMATH DE number 6837918