Free profinite semigroups over semidirect products
type: Item
instance of: scholarly article
title: Free profinite semigroups over semidirect products
zbMATH Open document ID: 0847.20055
published in: Russian Mathematics
publication date: 1996-05-21T00:00:00Z
review text: Free profinite semigroups and monoids over semidirect products are studied in the paper. A general description of these free objects over profinite spaces of generators is given in section 2. In case of semidirect products of the form \(J_1*V\) (where \(J_1\) is the pseudovariety of finite semilattice monoids) the description is interpreted in terms of profinite graphs (Section 3); also analogous results for the two-sided semidirect products are given in Section 5. Using this the authors make a complete study of the pseudovariety \(J_1*\text{Nil}\) (\(\text{Nil}\) is the pseudovariety of the finite nilpotent semigroups), i.e. give pseudoidentities defining \(J_1*\text{Nil}\), give a description of the corresponding free profinite semigroups, solve the word problem for them, give a description of the variety of rational languages associated with \(J_1*\text{Nil}\) (Section 4). In section 6 the authors construct the free profinite inverse monoid on a profinite alphabet \(A\) and show that the free profinite monoid over \(J_1*G\) (\(G\) is the pseudovariety of all finite groups) on \(A\) can be identified with one of its closed submonoids. Both these profinite monoids are described in terms of the Cayley graphs of the free profinite group.
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 08C15
zbMATH DE Number: 862274
zbMATH Keywords: semidirect products
zbMATH Keywords: free objects over profinite spaces of generators
zbMATH Keywords: profinite graphs
zbMATH Keywords: pseudovarieties
zbMATH Keywords: pseudoidentities
zbMATH Keywords: free profinite semigroups
zbMATH Keywords: word problem
zbMATH Keywords: variety of rational languages
zbMATH Keywords: free profinite inverse monoids
zbMATH Keywords: profinite alphabets
zbMATH Keywords: free profinite monoids
zbMATH Keywords: Cayley graphs
zbMATH Keywords: free profinite groups
author: Jorge Almeida
author: Pascal Weil
reviewed by: Pavel Zalesskij
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 862274