Free strict inverse semigroups
type: Item
instance of: scholarly article
title: Free strict inverse semigroups
zbMATH Open document ID: 0782.20047
author: K. Auinger
published in: Journal of Algebra
publication date: 1994-03-10T00:00:00Z
review text: An inverse semigroup is ``strict'' if it is a subdirect product of Brandt semigroups and groups. Strict inverse semigroups form a variety of inverse semigroups. Other than the varieties of Clifford semigroups, its subvarieties are those of the form \({\mathbf H}\vee{\mathbf B}\), where \(\mathbf H\) is some variety of groups and \(\mathbf B\) is the variety of combinatorial strict inverse semigroups, generated by the 5-element combinatorial Brandt semigroup. [See \textit{M. Petrich}, Inverse Semigroups (Wiley, 1984; Zbl 0546.20053).] The free strict inverse semigroups were first described by \textit{N. R. Reilly} [J. Lond. Math. Soc., II. Ser. 39, 102-120 (1989; Zbl 0636.20032)] and then by \textit{S. W. Margolis}, \textit{J. C. Meakin} and \textit{J. B. Stephen} [Can. J. Math. 42, 1084-1097 (1990; Zbl 0726.20040)]. The author offers an alternative structure theorem that is based on his work on the structure of semigroups whose congruence lattices satisfy certain conditions, using a construction of strict inverse semigroups in terms of the poset of \(\mathcal J\)-classes, the principal factors and certain partial homomorphisms between them (essentially equivalent to that of \textit{B. M. Schein} [see Petrich, above]). The free combinatorial strict inverse semigroup is explicitly described in these terms, those for the varieties \({\mathbf H}\vee{\mathbf B}\) less explicitly. In a sequel [J. Lond. Math. Soc., II. Ser. 45, 491-507 (1992; Zbl 0759.20016)] the author has refined his construction to provide an explicit one of the free strict inverse semigroup itself. The primary advantage of his method over the others is that his general structure theorems apply to wide classes of regular semigroups as well as inverse semigroups and can therefore be applied to the structure of various ``\(e\)-free'' and ``*-free'' regular semigroups.
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M18
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 202981
zbMATH Keywords: subdirect product
zbMATH Keywords: Brandt semigroups
zbMATH Keywords: variety of inverse semigroups
zbMATH Keywords: varieties of Clifford semigroups
zbMATH Keywords: variety of groups
zbMATH Keywords: variety of combinatorial strict inverse semigroups
zbMATH Keywords: 5-element combinatorial Brandt semigroup
zbMATH Keywords: free strict inverse semigroups
zbMATH Keywords: congruence lattices
zbMATH Keywords: regular semigroups
reviewed by: Peter R. Jones
MaRDI profile type: Publication
full work available at URL: jabr.1993.1088
OpenAlex ID: W2048168122
DOI: 10.1006/JABR.1993.1088
description: scientific article; zbMATH DE number 202981