Irreducible varieties of commutative semigroups
type: Item
instance of: scholarly article
title: Irreducible varieties of commutative semigroups
zbMATH Open document ID: 1026.20040
DOI: 10.1016/S0021-8693(02)00674-9
published in: Journal of Algebra
publication date: 2003-04-28T00:00:00Z
review text: This paper is based on a complete structural description of the lattice of varieties of commutative semigroups which is due to \textit{A. Kisielewicz} [Trans. Am. Math. Soc. 342, No. 1, 275-306 (1994; Zbl 0801.20042)]: each variety corresponds uniquely to a certain quadruple \((J,m,r,\pi)\), where \(\pi\) is a so-called `remainder' of the triple \((J,m,r)\). In a previous joint paper with \textit{A. Kisielewicz} [J. Algebra 232, No. 2, 493-506 (2000; Zbl 0970.20033)], the author has used this structural description to effectively characterize the covers and the dual covers of a given variety in the lattice. In the present paper, through a finer combinatorial analysis which leads to a proof of the existence of unique minimal generators for remainders, those characterizations are further refined by describing covers and dual covers in lattices of remainders. This allows the author to give a complete description of both join- and meet-irreducibles in the lattice of varieties of commutative semigroups. As a simple application, he deduces that the variety of all commutative semigroups is the only element of this lattice which is both join- and meet-irreducible.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 20M14
zbMATH DE Number: 1901961
zbMATH Keywords: commutative semigroups
zbMATH Keywords: lattices of varieties
zbMATH Keywords: irreducible elements
zbMATH Keywords: dual covers
author: Mariusz Grech
reviewed by: Jorge Almeida
MaRDI profile type: Publication
cites work: Some Order Properties of the Lattice of Varieties of Commutative Semigroups
cites work: Covering relation for equational theories of commutative semigroups
cites work: Varieties of Commutative Semigroups
cites work: Q3751631
cites work: Bases for equational theories of semi-groups
description: scientific article; zbMATH DE number 1901961