On transitive linear semigroups
type: Item
instance of: scholarly article
title: On transitive linear semigroups
zbMATH Open document ID: 0970.47024
DOI: 10.1016/S0024-3795(99)00229-3
author: Leo Livshits
author: Peter Šemrl
author: Roman Drnovšek
author: Gordon Macdonald
author: Ben Mathes
author: H. Radjavi
published in: Linear Algebra and its Applications
publication date: 2001-10-16T00:00:00Z
review text: A semigroup \({\mathcal S}\) of linear transformations acting on a finite-dimensional vector space \({\mathcal V}\) over a field \(\mathbb{F}\) acts transitively if for all \(x\neq 0\), \(y\in{\mathcal V}\) there exist \(S\in{\mathcal S}\) with \(Sx= y\). The action is sharply transitive if \(S\) is uniquely determined by \((x,y)\). The action is called pre-transitive if \({\mathcal S}(x)\supseteq{\mathcal V}\setminus\{0\}\) for all \(x\neq 0\). (Note that -- with this notation -- a group \({\mathcal S}\) cannot act transitively, but pre-transitively. And a sharply transitive semigroup \({\mathcal S}\) without zero is a group.) \({\mathcal S}\) is called left \(t\)-simple if the semigroup contains no proper transitive left ideal in \({\mathcal S}\). In the paper under review, the authors investigate transitive left \(t\)-simple semigroups (and some related classes) and characterize e.g. among others transitive semigroups which are representable as unions of groups and a zero element.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 20M30
zbMATH DE Number: 1419714
zbMATH Keywords: invariant subspace
zbMATH Keywords: irreducible \(t\)-simple semigroup
zbMATH Keywords: left \(t\)-simple semigroups
zbMATH Keywords: transitive semigroups
reviewed by: Wilfried Hazod
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1419714