The structure of \(\mathcal Q^*\)-inverse semigroups.
type: Item
instance of: scholarly article
title: The structure of \(\mathcal Q^*\)-inverse semigroups.
zbMATH Open document ID: 1219.20040
published in: Journal of Algebra
publication date: 2011-03-22T00:00:00Z
review text: A semigroup \(S\) is called abundant if every \(\mathcal L^*\)-class and every \(\mathcal R ^*\)-class contains at least one idempotent of \(S\). An abundant semigroup \(S\) is called idempotent connected (IC) if for each element \(a\in S\) and for some \(a^+\in R_a^*(S)\cap E\), \(a^*\in L_a^*(S)\cap E\) there exists a bijection \(\alpha\colon\langle a^+\rangle\to\langle a^*\rangle\) such that \(xa=a(x\alpha)\) for all \(x\in\langle a^+\rangle\). An IC abundant semigroup \(S\) is called \(\mathcal Q\)-inverse semigroup if the set of all its idempotents \(E\) forms a regular band, that is, \(efege=efge\) for all \(e,f,g\in E\). Thus, the class of \(\mathcal Q\)-inverse semigroups is a generalization of the class of quasi-inverse semigroups. A construction theorem of \(\mathcal Q\)-inverse semigroups is given by using the wreath product of some semigroups. It is proved that a semigroup \(S\) is a \(\mathcal Q\)-inverse semigroup if and only if \(S\) is a spined product of an \(\mathcal L^*\)-inverse semigroup and an \(\mathcal R^*\)-inverse semigroup. Here a spined product of semigroups \(S_1\) and \(S_2\) is a semigroup \(S\) defined by means of two homomorphisms \(\theta_1\colon S_1\to H\) and \(\theta_2\colon S_2\to H\) onto their common homomorphic image \(H\): \(S=\{(s_1,s_2)\in S_1\times S_2\mid s_1\theta_1=s_2\theta_2\}\).
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M18
Mathematics Subject Classification ID: 20M19
zbMATH DE Number: 5869002
zbMATH Keywords: quasi-inverse semigroups
zbMATH Keywords: \(\mathcal L\)-inverse semigroups
zbMATH Keywords: \(\mathcal Q\)-inverse semigroups
zbMATH Keywords: wreath products
zbMATH Keywords: spined products
zbMATH Keywords: type A semigroups
zbMATH Keywords: abundant semigroups
zbMATH Keywords: idempotents
author: Xueming Ren
author: K. P. Shum
reviewed by: Grigori I. Zhitomirski
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.09.020
OpenAlex ID: W2095282204
cites work: Structure of concordant semigroups
cites work: The structure of type A semigroups
cites work: Right inverse semigroups
cites work: Idempotent-connected abundant semigroups
cites work: Quasi-adequate semigroups
cites work: Quasi-adequate semigroups. II
cites work: Adequate Semigroups
cites work: Abundant Semigroups
cites work: Q4262339
cites work: On regular semigroups
cites work: Q4126563
cites work: Lattice isomorphisms of inverse semigroups
cites work: THE STRUCTURE OF TYPE A SEMIGROUPS
cites work: The natural partial order on an abundant semigroup
cites work: Good congruences on a type σ semigroup*
cites work: One-to-one partial right translations of a right cancellative semigroup
cites work: A representation of a semigroup by a semigroup of matrices over a group with zero
cites work: Semidirect products of semigroups
cites work: Q4484922
cites work: The structure of superabundant semigroups.
cites work: The structure of \(\mathcal L^*\)-inverse semigroups.
cites work: Right (left) inverse semigroups
cites work: Orthodox semigroups whose idempotents satisfy a certain identity
cites work: Note on a certain class of orthodox semigroups
DOI: 10.1016/J.JALGEBRA.2010.09.020
description: scientific article; zbMATH DE number 5869002