Congruence relations on the lattice of existence varieties of regular semigroups
type: Item
instance of: scholarly article
title: Congruence relations on the lattice of existence varieties of regular semigroups
zbMATH Open document ID: 0842.20051
author: Norman R. Reilly
author: Shuhua Zhang
published in: Journal of Algebra
publication date: 1996-02-15T00:00:00Z
review text: A class \(\mathcal V\) of regular semigroups is an existence variety (or e-variety) if it is closed under the formation of direct products, regular subsemigroups and homomorphic images. The collection of all e-varieties forms a complete lattice \({\mathcal L}_{ev}\) under inclusion which lattice is the topic of the paper under review. The general line of this investigation is via complete congruences on \({\mathcal L}_{ev}\) which decompose the lattice into disjoint bounded intervals, and idempotent operators on the lattice associated with the congruences. The method of how such complete congruences are found -- radical congruence systems on the underlying class of semigroups -- is merely of universal algebraic nature and is roughly described as follows. Let \(\mathcal C\) be a class of universal algebras; a congruence system \(\rho = \{\rho_S \mid S \in {\mathcal C}\}\) assigns to each member \(S\) of \(\mathcal C\) a congruence \(\rho_S\) on \(S\) in a ``natural way''. (For example, if \(\mathcal C\) consists of all regular semigroups or of all finite semigroups then a typical congruence system is \(\mu\) which assigns to each such semigroup \(S\) the greatest idempotent separating congruence \(\mu_S\) on \(S\).) If \(\mathcal C\) is a variety of universal algebras or the e-variety of all regular semigroups then two types of congruence systems which satisfy some further compatibility conditions are of particular importance (and are called ``(e-)varietal radical congruence system'', respectively ``HS-compatible radical congruence system''). The importance comes from the fact that each such system leads naturally to a complete congruence on the lattice of all sub-(e-)varieties of \(\mathcal C\). (The construction has its origin in some earlier work of Petrich and Reilly concerning the lattice of varieties of completely regular semigroups [\textit{M. Petrich} and \textit{N. R. Reilly}, J. Algebra 134, No. 1, 1-27 (1990; Zbl 0706.20043)].) It is important to note that…
reviewed by: K. Auinger
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 840727
zbMATH Keywords: existence variety
zbMATH Keywords: complete congruences
zbMATH Keywords: radical congruence systems
zbMATH Keywords: regular semigroups
zbMATH Keywords: idempotent separating congruences
zbMATH Keywords: lattice of varieties
MaRDI profile type: Publication
full work available at URL: jabr.1995.1375
OpenAlex ID: W2083616227
DOI: 10.1006/JABR.1995.1375
Recommended article: Relations on the lattice of congruences of a regular semigroup
Recommended article: Certain relations on the congruence lattice of a completely regular semigroup
Recommended article: Q3778168
Recommended article: Certain Relations on a Lattice of Varieties of Completely Regular Semigroups
Recommended article: Relations on a lattice of varieties of completely regular semigroups
Recommended article: The congruence lattice of a regular semigroup
Recommended article: Q5185093
Recommended article: Q3767513
Recommended article: Q3322298
Recommended article: Q5696832
description: scientific article; zbMATH DE number 840727