Free profinite \(\mathcal R\)-trivial, locally idempotent and locally commutative semigroups
type: Item
instance of: scholarly article
title: Free profinite \(\mathcal R\)-trivial, locally idempotent and locally commutative semigroups
zbMATH Open document ID: 0939.20057
DOI: 10.1007/BF03325439
author: José Carlos Costa
published in: Semigroup Forum
publication date: 1999-11-16T00:00:00Z
review text: For each of the Green relations \({\mathcal T}\in\{\mathcal{R,L,J}\}\), let \(\mathbf T\) denote the pseudovariety of all finite \(\mathcal T\)-trivial semigroups. Let also \(\mathbf{DA}\) denote the pseudovariety of all finite semigroups in which regular elements are idempotents and \(\mathbf{LJ}_1\) the pseudovariety of all finite semigroups which are locally semilattices. The central results in the paper describe the finitely generated free profinite semigroups over the intersection \({\mathbf R}\cap\mathbf{LJ}_1\). For these semigroups, a normal factorization in terms of generators and idempotents is given, idempotents being characterized by means of their content and their restriction to semigroups in which idempotents are left zeros. These results yield corresponding results for free profinite semigroups over \({\mathbf J}\cap\mathbf{LJ}_1\), which reproves results of \textit{C.~Selmi} [Theor. Comput. Sci. 161, No. 1-2, 157-190 (1996; Zbl 0872.68092)], and \(\mathbf{DA}\cap\mathbf{LJ}_1\), and allow the author to prove that the latter intersection is the join of \({\mathbf R}\cap\mathbf{LJ}_1\) with \({\mathbf L}\cap\mathbf{LJ}_1\). Further applications are obtained by interpreting these results and their proofs in terms of varieties of rational languages. (Also submitted to MR).
reviewed by: Jorge Almeida
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M35
zbMATH DE Number: 1309716
zbMATH Keywords: free profinite semigroups
zbMATH Keywords: pseudovarieties of finite semigroups
zbMATH Keywords: varieties of rational languages
zbMATH Keywords: Green relations
zbMATH Keywords: generators
zbMATH Keywords: idempotents
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1309716