On special types of Putcha semigroups whose subgroups belong to a given variety of groups.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1258.20045
publication date: 2012-07-20T00:00:00Z
full work available at URL: index.html
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M11
zbMATH DE Number: 6058657
zbMATH Keywords: semilattices of Archimedean semigroups
zbMATH Keywords: Putcha semigroups
zbMATH Keywords: semilattice congruences
zbMATH Keywords: Rhodes radical
zbMATH Keywords: idempotents
zbMATH Keywords: pseudovarieties of groups
MaRDI profile type: Publication
author: Attila Nagy
title: On special types of Putcha semigroups whose subgroups belong to a given variety of groups.
published in: International Journal of Algebra
review text: A semigroup such that \(a\) divides \(b\) implies \(a^2\) divides some power of \(b\) is said to be a `Putcha semigroup' in honor of \textit{M. S. Putcha} [Semigroup Forum 6, 12-34 (1973; Zbl 0256.20074)], who showed that this property characterizes the semilattices of Archimedean semigroups. Given a variety \(\mathbf H\) of groups, a congruence on a semigroup is said to be an `\(\mathbf{LH}\)-congruence' if the local submonoids of every idempotent class are groups from \(\mathbf H\).NEWLINENEWLINE The main result in the paper states that, for a semilattice of ideal extensions of completely simple semigroups, the subgroups belong to \(\mathbf H\) if and only if the least semilattice congruence is a maximal \(\mathbf{LH}\)-congruence. This result applies in particular to periodic Putcha semigroups with only finitely many idempotents and to periodic permutative semigroups. In the finite case, the author also identifies the least semilattice congruence of a Putcha semigroup with its Rhodes radical, using a result of the reviewer et al. [Trans. Am. Math. Soc. 361, No. 3, 1429-1461 (2009; Zbl 1185.20058)].
reviewed by: Jorge Almeida
description: scientific article; zbMATH DE number 6058657