Idempotent pure extensions by inverse semigroups via quivers
type: Item
instance of: scholarly article
title: Idempotent pure extensions by inverse semigroups via quivers
zbMATH Open document ID: 0934.20045
DOI: 10.1016/S0022-4049(97)00110-2
author: Gracinda M. S. Gomes
author: Mária B. Szendrei
published in: Journal of Pure and Applied Algebra
publication date: 2000-04-06T00:00:00Z
review text: A quiver consists of a set of objects and, for any objects \(u\) and \(v\), of a set of arrows from \(u\) to \(v\). On the set of arrows a partial binary operation \(+\) is defined and this yields a structure which is like a category except that identities need not exist, and addition of non-consecutive arrows is possible. Given an inverse monoid \(T\) and a partial action of \(T\) on a quiver \(\mathcal C\) which has object set \(T\), the authors define the notion of an inverse \(T\)-quiver. To each inverse \(T\)-quiver \(\mathcal C\) the authors associate in a canonical way an inverse semigroup \(S({\mathcal C})\). If in particular for every object \(a\) of \(\mathcal C\) the set of arrows from \(a\) to \(a\) constitutes a semilattice for the operation \(+\), then there exists a naturally defined idempotent pure congruence \(\delta\) on \(S({\mathcal C})\) such that \(S({\mathcal C})/\delta\) is embeddable into \(T\), i.e., \(S({\mathcal C})\) is an idempotent pure extension by an inverse subsemigroup of \(T\). Conversely, if \(S\) is an inverse semigroup which is an idempotent pure extension by an inverse senigroup \(T\), then a so-called derived quiver of the extension is constructed. It is proven that this derived quiver \(\mathcal D\) is a \(T^1\)-quiver with \(S=S({\mathcal D})\). The paper contains several variants of these results. In the last two sections, results by O'Carroll and Billhardt concerning idempotent pure extensions by inverse semigroups are derived.
Mathematics Subject Classification ID: 20M18
Mathematics Subject Classification ID: 20M17
zbMATH DE Number: 1308155
zbMATH Keywords: inverse monoids
zbMATH Keywords: partial actions
zbMATH Keywords: inverse semigroups
zbMATH Keywords: idempotent pure congruences
zbMATH Keywords: idempotent pure extensions
zbMATH Keywords: derived quivers
reviewed by: Francis Pastijn
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308155