Abundant semigroups whose idempotents satisfy permutation identities
type: Item
instance of: scholarly article
title: Abundant semigroups whose idempotents satisfy permutation identities
zbMATH Open document ID: 0885.20038
DOI: 10.1007/BF02676614
published in: Semigroup Forum
publication date: 1998-02-17T00:00:00Z
full work available at URL: https://eudml.org/doc/135531
review text: Let \(S\) be a semigroup and \(a,b\in S\). Put \(a{\mathcal L}^*b\) (respectively, \(a{\mathcal R}^*b\)) whenever \(a\) and \(b\) are related by Green's relation \(\mathcal L\) (respectively, \(\mathcal R\)) in some oversemigroup of \(S\). A semigroup is called abundant if each \({\mathcal L}^*\)-class and each \({\mathcal R}^*\)-class contains an idempotent. An abundant semigroup \(S\) is called quasi-adequate if its idempotents form a subsemigroup of \(S\); if, moreover, idempotents of \(S\) permute then \(S\) is called adequate. The set of idempotents of \(S\) is denoted by \(E(S)\). Theorem 2.1 states that, for an abundant semigroup \(S\), the following are equivalent: (1) \(E(S)\) satisfies a permutation identity; (2) \(E(S)\) is a normal band; (3) \(S\) is quasi-adequate and \(eSe\) is an adequate subsemigroup of \(S\) for any \(e\in E(S)\). Let \(T\) be an adequate semigroup with semilattice of idempotents \(Y\), and let \(L=S(Y;L_a;\varphi_{a,b})\) (respectively \(R=S(Y;R_a;\psi_{a,b})\)) be a strong semilattice of left (right) zero bands \(L_a\) (respectively, \(R_a\)). The author construct a new semigroup depending on \(L,R,T\) and \(Y\), call it the quasi-spined product of \(L,R\) and \(T\) on \(Y\) and denote it by \(QS(L,R,T;Y)\). According to Theorem 2.3, if \(S=QS(L,R,T;Y)\) then \(S\) is an abundant semigroup whose idempotents satisfy a permutation identity and \(S/\delta\cong T\) for some natural congruence \(\delta\) on \(S\). Conversely, Theorem 2.4 states that any abundant semigroup whose idempotents satisfy a permutation identity is isomorphic to some semigroup of the kind \(QS(L,R,T;Y)\). Theorem 3.4 states that, for an abundant semigroup \(S\), the following are equivalent: (1) \(S\) satisfies a permutation identity; (2) \(S\) satisfies the identity \(xyzt=xzyt\); (3) \(S\) is isomorphic to a spined product of a normal band and a commutative semigroup which is a semilattice of cancellative monoids. (The definition of the spined product see, fo…
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 1022339
zbMATH Keywords: abundant semigroups
zbMATH Keywords: adequate semigroups
zbMATH Keywords: strong semilattices of bands
zbMATH Keywords: permutation identities
zbMATH Keywords: quasi-spined products
zbMATH Keywords: semilattices of idempotents
author: Xiaojiang Guo
reviewed by: Boris Vernikov
MaRDI profile type: Publication
cites work: Structure of concordant semigroups
cites work: Quasi-adequate semigroups
cites work: Abundant Semigroups
cites work: Adequate Semigroups
cites work: Q4126563
cites work: Q4148155
cites work: Regular semigroups whose idempotents satisfy permutation identities
description: scientific article; zbMATH DE number 1022339