Inverse semigroups and extensions of groups by semilattices
type: Item
instance of: scholarly article
title: Inverse semigroups and extensions of groups by semilattices
zbMATH Open document ID: 0625.20043
DOI: 10.1016/0021-8693(87)90046-9
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: This is the first of three papers on inverse semigroups published by the authors in J. Algebra [see the following reviews]. Here they generalize the theory of P-semigroups (which characterize E-unitary inverse semigroups) to the non-regular case. The main theorem, from which the McAlister P-Theorem is recovered, is as follows: Let M be a monoid whose idempotents commute. The following are equivalent: (1) There is a surjective morphism \(\phi\) : \(M\to G\) onto a group such that \(1\phi^{-1}=E(M)\). (2) M is E-unitary and dense (meaning that for any \(s\in M\) there exists \(s_ 1,s_ 2\) such that \(ss_ 1,s_ 2s\in E(M))\). (3) M is isomorphic to a monoid C/G where G is a group acting transitively without fixpoints on a connected, idempotent and commutative category C. The group G can be taken as the ``free group over M'' and C the corresponding ``derived covering'': all these terms are defined and developed within the paper.
reviewed by: Peter M. Higgins
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M50
zbMATH DE Number: 4015008
zbMATH Keywords: extensions of groups by semilattices
zbMATH Keywords: inverse semigroups
zbMATH Keywords: P-semigroups
zbMATH Keywords: E-unitary inverse semigroups
zbMATH Keywords: P-Theorem
zbMATH Keywords: idempotents
author: S. W. Margolis
author: Jean-Éric Pin
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90046-9
OpenAlex ID: W2006765004
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description: scientific article; zbMATH DE number 4015008