Operators and products in the lattice of existence varieties of regular semigroups
type: Item
instance of: scholarly article
title: Operators and products in the lattice of existence varieties of regular semigroups
zbMATH Open document ID: 0854.20069
DOI: 10.1007/BF02574117
author: Norman R. Reilly
author: Shuhua Zhang
published in: Semigroup Forum
publication date: 1996-12-01T00:00:00Z
full work available at URL: https://eudml.org/doc/135466
review text: The authors introduced several complete congruences on the lattice \(L={\mathcal L}_{ev}({\mathcal R}{\mathcal S})\) of e- (or ``existence'') varieties of regular semigroups [in J. Algebra 178, No. 3, 733-759 (1995; Zbl 0842.20051)], generalizing the corresponding notions for the lattice of varieties of completely regular semigroups. Whilst in the latter case these congruences have natural syntactic descriptions, this cannot be so in general, for free regular semigroups do not exist. Thus alternative interpretations are desirable. Since the cited paper was written, various authors, including K. Auinger and T. E. Hall, have in fact generated further such congruences, together with some such interpretations. The purpose of the current paper is to provide descriptions of four of the original congruences in terms of semidirect products. Given any complete congruence \(\rho\) on the lattice \(L\), and an e-variety \(\mathcal U\), the \(\rho\)-class \({\mathcal U}\rho\) is an interval, of the form \([{\mathcal U}_\rho,{\mathcal U}^\rho]\). For \(\rho=T_l\) or \(K_l\), \({\mathcal U}^\rho\) is [loc. cit.] the Malcev product \({\mathcal L}{\mathcal G}\circ{\mathcal U}\) or \({\mathcal L}{\mathcal Z}\circ{\mathcal U}\), respectively, where \({\mathcal L}{\mathcal G}\) and \({\mathcal L}{\mathcal Z}\) are the e-varieties of left groups and of left zero semigroups, respectively. There are natural duals. (Alternatively, these formulas may be used to define the respective congruences). The relationship between Malcev products and semidirect products has a long history for varieties of groups and for varieties, and pseudovarieties, of semigroups. Thus it is natural to look for descriptions in terms of semidirect (or wreath) products. One of the main results of the paper is that for any nongroup e-variety \(\mathcal U\), \({\mathcal U}^{T_l}={\mathcal G}*{\mathcal U}\), where \({\mathcal G}\) is the (e-) variety of groups and \({\mathcal G}*{\mathcal U}\) is the e-variety generate…
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 895847
zbMATH Keywords: e-varieties of regular semigroups
zbMATH Keywords: wreath products
zbMATH Keywords: complete congruences
zbMATH Keywords: lattice of varieties of completely regular semigroups
zbMATH Keywords: semidirect products
zbMATH Keywords: Malcev product
zbMATH Keywords: e-varieties of left groups
zbMATH Keywords: left zero semigroups
zbMATH Keywords: varieties of groups
zbMATH Keywords: pseudovarieties
zbMATH Keywords: completely simple semigroups
reviewed by: Peter R. Jones
MaRDI profile type: Publication
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OpenAlex ID: W1965816059
description: scientific article; zbMATH DE number 895847