Congruences associated with inverse transversals
type: Item
instance of: scholarly article
title: Congruences associated with inverse transversals
zbMATH Open document ID: 0849.20042
author: T. S. Blyth
author: M. H. Almeida Santos
published in: Collectanea Mathematica
publication date: 1996-10-27T00:00:00Z
full work available at URL: https://eudml.org/doc/39995
review text: An inverse transversal of a regular semigroup \(S\) is a regular subsemigroup \(S^\circ\) such that \(S^\circ\) contains a unique inverse \(x^\circ\) of every \(x\in S\). The properties of the congruence lattices \(\overline{\text{Con }}S\) of the algebra \((S,\cdot,{}^\circ)\) and the properties of the lattice \(\text{Con }S\) of congruences on \(S\) are considered. An inverse transversal \(S^\circ\) is said to be multiplicative if \(\Lambda I=E(S^\circ)\), where \(E(S^\circ)\) is the set of idempotents of \(S^\circ\), \(\Lambda=\{x^\circ x, x\in S\}\) and \(I=\{x x^\circ, x\in S\}\). If the inverse transversal \(S^\circ\) is multiplicative, then \(\overline{\text{Con}} S\overline{\sim}\text{Con }S(x)\),where \(\sim\) is the dual closure equivalence defined by \(\vartheta\sim\varphi\Leftrightarrow\vartheta|_I=\varphi|_I\), \(\vartheta|_\Lambda=\varphi|_\Lambda\), \(\vartheta|_{S^\circ}=\varphi|_{S^\circ}\).
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 869942
zbMATH Keywords: inverse transversals
zbMATH Keywords: regular semigroups
zbMATH Keywords: congruence lattices
zbMATH Keywords: idempotents
zbMATH Keywords: dual closure equivalence
reviewed by: Jaak Henno
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 869942