Minimal group-universal varieties of semigroups
type: Item
instance of: scholarly article
title: Minimal group-universal varieties of semigroups
zbMATH Open document ID: 0591.20058
DOI: 10.1007/BF01187561
author: Pavel Goralčík
author: Václav Koubek
published in: Algebra Universalis
publication date: 1985-01-01T00:00:00Z
review text: A category \({\mathcal C}\) is ''group-universal'' if any group G is isomorphic to Aut X for some object X in \({\mathcal C}\), (in which case X is said to represent G). For instance group-universality of graphs was proved by \textit{R. Frucht} [Compos. Math. 6, 239-250 (1938; Zbl 0020.07804)]. Group- universality of semigroups (in fact commutative semigroups) was proven by \textit{Z. Hedrlín} and \textit{J. Lambek} [J. Algebra 11, 195-212 (1969; Zbl 0206.025)]. The authors determine here the minimal group-universal varieties of semigroups. They are (1) the variety of semilattices; (2) the variety \({\mathcal M}\) generated by a certain four-element semigroup; (3) the variety of all inflations of rectangular bands; (4) the varieties \(\sqrt{{\mathcal L}}\) and \(\sqrt{{\mathcal R}}\), consisting of all semigroups S for which \(S^ 2\) is left, respectively right, zero; and for each prime p, (5) the variety of all inflations of elementary abelian groups of exponent p, and (6) the variety generated by the \(2\times 2\) nonorthodox Rees matrix semigroup over the cyclic group of order p. In each case a representing semigroup for a given group is constructed via a representing graph. This yields bounds on the cardinalities required.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 3948609
zbMATH Keywords: minimal group-universal varieties of semigroups
zbMATH Keywords: variety of semilattices
zbMATH Keywords: inflations of rectangular bands
Wikidata QID: Q115943454
reviewed by: Peter R. Jones
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01187561
OpenAlex ID: W2010423711
description: scientific article; zbMATH DE number 3948609