Congruences on strong semilattices of regular simple semigroups
type: Item
instance of: scholarly article
title: Congruences on strong semilattices of regular simple semigroups
zbMATH Open document ID: 0643.20041
DOI: 10.1007/BF02573132
published in: Semigroup Forum
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134923
review text: In this thesis [Univ. Wien, 1982; Zbl 0474.20035] \textit{K. Auinger} studied congruences on strong semilattices Y of regular and simple semigroups \(S_{\alpha}\) (\(\alpha\in Y)\). In this paper the author adopts a similar development for the construction of all congruences considering congruences on Y and for each \(\alpha\in Y\) a congruence on \(S_{\alpha}\) satisfying certain simple compatibility conditions. (This development does not make use of the kernel-trace approach introduced by \textit{F. Pastijn} and \textit{M. Petrich} [in Trans. Am. Math. Soc. 295, 607- 633 (1986; Zbl 0599.20095)].) First, a characterization of such semigroups in terms of the natural partial order on a regular semigroup is given. Using the main characterization theorem for congruences on strong semilattices of regular and simple semigroups, S, the special case is studied, where all the linking homomorphisms different from the identity are constant. Also, the smallest simple congruence \(\theta\) on such semigroups is described. Necessary and sufficient conditions are found in order that the lattice of congruences C(S) be semi-modular, M- symmetric, modular or distributive: C(S) has any of these properties iff C(Y) and \(C(S_{\alpha})\) for all \(\alpha\in Y\) have the same property and all the linking homomorphisms are constant. Finally, complementation in C(S) is investigated, together with the Boolean case (see also K. Auinger, loc. cit.): C(S) is complemented iff S is a subdirect product of S/J and S/\(\theta\) and both C(S/J) and C(S/\(\theta)\) are complemented.
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 4047965
zbMATH Keywords: congruences on strong semilattices of regular and simple semigroups
zbMATH Keywords: lattice of congruences
zbMATH Keywords: semi-modular
zbMATH Keywords: distributive
zbMATH Keywords: complementation
author: Mario Petrich
reviewed by: H. Mitsch
MaRDI profile type: Publication
cites work: Semimodularity in lattices of congruences
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cites work: Semimodular Inverse Semigroups
cites work: On congruence lattices of regular semigroups
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cites work: A Natural Partial Order for Semigroups
cites work: Q4083905
OpenAlex ID: W2039444583
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description: scientific article; zbMATH DE number 4047965