The bifree locally inverse semigroup on a set
type: Item
instance of: scholarly article
title: The bifree locally inverse semigroup on a set
zbMATH Open document ID: 0806.20052
published in: Journal of Algebra
publication date: 1994-08-17T00:00:00Z
review text: A `locally inverse' semigroup is a regular semigroup \(S\) all of whose local submonoids (those of the form \(eSe\), for \(e \in E(S)\), its set of idempotents) are inverse. The class \({\mathcal L}{\mathcal I}\) of locally inverse semigroups is closed under products, homomorphic images and regular subsemigroups and as such forms an existence (or \(e\)-) variety. It was shown by \textit{Y. T. Yeh} [Int. J. Algebra Comput. 2, 471-484 (1992; Zbl 0765.20030)] that every \(e\)-variety \(\mathcal V\) of locally inverse semigroups possesses a `bifree object' \(BF{\mathcal V}(X)\) for each nonempty set \(X\), defined by the property that every `matched' mapping \(\phi\) from \(X \cup X'\) (where \(X'\) is a set of formal inverses for the elements of \(X\) and `matched' requires that \(x'\phi\) be an inverse of \(x \phi\) in \(S\) for every \(x\) in \(X\)) to a member \(S\) of \(\mathcal V\) extends uniquely to a morphism from \(BF{\mathcal V}(X)\) to \(S\). The purpose of the current paper is to describe \(BF{\mathcal L}{\mathcal I}(X)\). By a result of \textit{F. Pastijn} [Trans. Am. Math. Soc. 273, 631-655 (1982; Zbl 0512.20042)], every locally inverse semigroup is a homomorphic image of a regular subsemigroup of a ``semidirect product of a semilattice by a completely simple semigroup''. The construction of the latter generalizes both the Rees construction and the McAlister construction of `\(P\)-semigroups'. It follows that such semigroups generate \(\mathcal L \mathcal I\) as an \(e\)-variety and that \(BF{\mathcal L}{\mathcal I})(X)\) is embedded in a semigroup of this type. As a necessary and interesting preliminary, the author first provides a Rees representation for the bifree completely simple semigroup \(BF{\mathcal C}{\mathcal S}(X)\) on \(X\). The `semidirect product' of a certain free semilattice with this completely simple semigroup is then constructed; we denote it by \(P\), for convenience. To identify the requisite subsemigroup, a further notion is needed.…
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M18
zbMATH DE Number: 614247
zbMATH Keywords: existence variety
zbMATH Keywords: regular semigroup
zbMATH Keywords: local submonoids
zbMATH Keywords: idempotents
zbMATH Keywords: locally inverse semigroups
zbMATH Keywords: \(e\)-variety
zbMATH Keywords: completely simple semigroup
zbMATH Keywords: bifree completely simple semigroup
zbMATH Keywords: free semilattice
zbMATH Keywords: word problem
zbMATH Keywords: membership
author: K. Auinger
reviewed by: Peter R. Jones
MaRDI profile type: Publication
full work available at URL: jabr.1994.1169
OpenAlex ID: W2044044495
DOI: 10.1006/JABR.1994.1169
description: scientific article; zbMATH DE number 614247