Commutativity of operators on the lattice of existence varieties
type: Item
instance of: scholarly article
title: Commutativity of operators on the lattice of existence varieties
zbMATH Open document ID: 0870.20042
DOI: 10.1007/BF01326768
author: Norman R. Reilly
author: Shuhua Zhang
published in: Monatshefte für Mathematik
publication date: 1997-08-28T00:00:00Z
full work available at URL: https://eudml.org/doc/178778
review text: A class of regular semigroups is called an e-variety if it is closed under direct products, homomorphic images and regular subsemigroups. The article continues earlier work of the authors [J. Algebra 178, No. 3, 733-759 (1995; Zbl 0842.20051); Semigroup Forum 53, No. 1, 1-24 (1996; Zbl 0854.20069)]. In the former article, the authors introduced 7 complete congruences on the lattice \({\mathbf L}\) of e-varieties of regular semigroups. In the work under review, 4 new complete congruences on \({\mathbf L}\) are introduced. They have the form \(\alpha_{\mathcal P}\): \(\mathcal A\alpha_{\mathcal P}\mathcal B\Leftrightarrow\mathcal A\cap\mathcal P=\mathcal B\cap\mathcal P\) where \(\mathcal P\) is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. To each complete congruence \(\rho\) on \({\mathbf L}\) is associated an idempotent operator \({\mathcal A}\rightarrow{\mathcal A}^{\rho}\) on \({\mathbf L}\). Numerous results concerning the commutativity of such operators are established.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 1019727
zbMATH Keywords: regular semigroups
zbMATH Keywords: lattices of e-varieties
zbMATH Keywords: complete congruences on lattices
zbMATH Keywords: idempotent operators
zbMATH Keywords: commutativity
reviewed by: Boris Vernikov
MaRDI profile type: Publication
cites work: The word problem for the bifree combinatorial strict regular semigroup
cites work: Bifree objects in e-varieties of strict orthodox semigroups and the lattice of strict orthodox *-semigroup varieties
cites work: Free strict inverse semigroups
cites work: The bifree locally inverse semigroup on a set
cites work: Q3848243
cites work: Q4840263
cites work: On regular semigroups whose idempotents form a subsemigroup
cites work: On regular semigroups
cites work: Some properties of local subsemigroups inherited by larger subsemigroups
cites work: Identities for existence varieties of regular semigroups
cites work: Regular semigroups: Amalgamation and the lattice of existence varieties
cites work: On the lattice of varieties of bands of groups
cites work: Q4126563
cites work: Q3322298
cites work: Q3754190
cites work: Semidirect products of regular semigroups
cites work: A new approach in the theory of orthodox semigroups
cites work: E-unitary inverse monoids and the Cayley graph of a group presentation
cites work: Q3895689
cites work: Congruences on Regular Semigroups
cites work: The congruence lattice of a regular semigroup
cites work: Q4021416
cites work: Residual finiteness in completely regular semigroup varieties
cites work: Q3337662
cites work: Near varieties of idempotent generated completely simple semigroups
cites work: Semigroups generated by certain operators on varieties of completely regular semigroups
cites work: Q3490195
cites work: Q3689372
cites work: Free Combinatorial Strict Inverse Semigroups
cites work: Congruences on regular semigroups
cites work: Congruence relations on the lattice of existence varieties of regular semigroups
cites work: Operators and products in the lattice of existence varieties of regular semigroups
cites work: Associativity of products of existence varieties of regular semigroups
cites work: Congruence extensions in regular semigroups
cites work: Q3890901
cites work: THE EXISTENCE OF E-FREE OBJECTS IN E-VARIETIES OF REGULAR SEMIGROUPS
cites work: On existence varieties of \(E\)-solid or locally inverse semigroups and \(E\)-invariant congruences
cites work: Completely regular semigroup varieties generated by Mal'cev products with groups
cites work: Certain operators related to Mal'cev products on varieties of completely regular semigroups
cites work: An infinite order operator on the lattice of varieties of completely regular semigroups
OpenAlex ID: W2037762010
description: scientific article; zbMATH DE number 1019727