Abelian pointlikes of a monoid
type: Item
instance of: scholarly article
title: Abelian pointlikes of a monoid
zbMATH Open document ID: 0909.20050
DOI: 10.1007/PL00005951
author: Manuel Delgado
published in: Semigroup Forum
publication date: 1999-04-07T00:00:00Z
review text: The ``Type II'' kernel \(K(M)\) of a finite monoid \(M\) comprises those elements of \(M\) that lie in the inverse image of the identity element, under every relational morphism from \(M\) into a finite group. The kernel plays an important role in the study of semidirect products and Malcev products of pseudovarieties of monoids (and semigroups in general). The ``Rhodes conjecture'' stated that membership in \(K(M)\) is determined by quite a simple algorithm. It was first proved by \textit{C. J. Ash} [Int. J. Algebra Comput. 1, No. 1, 127-146 (1991; Zbl 0722.20039)] and independently by \textit{L. Ribes} and \textit{P. A. Zalesskij} [Bull. Lond. Math. Soc. 25, No. 1, 37-43 (1993; Zbl 0811.20026)], the latter using the methods of profinite group theory. The author considers the abelian kernel of a finite monoid -- where only relational morphisms into abelian groups are allowed -- by studying the profinite topology of free abelian groups. Together with a little linear algebra, this work enables him to find an algorithm to determine membership in the abelian kernel and to obtain some information about the pointlikes with respect to such morphisms (a pointlike set must be contained in the inverse image of a single element, for every relational morphism into an abelian group).
Mathematics Subject Classification ID: 20M35
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 1156727
zbMATH Keywords: finite monoids
zbMATH Keywords: kernels
zbMATH Keywords: semidirect products
zbMATH Keywords: Malcev products
zbMATH Keywords: pseudovarieties of monoids
zbMATH Keywords: abelian kernels
zbMATH Keywords: relational morphisms
zbMATH Keywords: profinite topology of free Abelian groups
zbMATH Keywords: algorithms
zbMATH Keywords: membership problem
zbMATH Keywords: pointlikes
reviewed by: Peter R. Jones
MaRDI profile type: Publication
full work available at URL: pl00005951
OpenAlex ID: W2085717272
description: scientific article; zbMATH DE number 1156727