Malcev presentations for subsemigroups of direct products of coherent groups.
type: Item
instance of: scholarly article
title: Malcev presentations for subsemigroups of direct products of coherent groups.
zbMATH Open document ID: 1178.20050
author: Alan J. Cain
published in: Journal of Pure and Applied Algebra
publication date: 2009-03-30T00:00:00Z
review text: Let \(\rho\) be a binary relation on the free semigroup \(A^+\) over an alphabet \(A\). Let \(\rho^M\) be the Malcev congruence generated by \(\rho\), that is, \(\rho\) is the smallest congruence \(\sigma\) such that \(A^+/\sigma\) is embeddable in a group. A semigroup is Malcev coherent if every of its finitely generated subsemigroups \(S\) has a finite Malcev presentation, that is, \(S\cong A^+/\rho^M\) with finite \(A\) and \(\rho\). The author proves that the direct product of two free semigroups of rank at least 2 is not Malcev coherent. This contrasts with the result by \textit{J. C. Spehner} [J. Pure Appl. Algebra 58, No. 3, 279-287 (1989; Zbl 0677.20046)] that free semigroups are Malcev coherent. As a byproduct he shows that polycyclic groups are not Malcev coherent. He also proves that the direct product of a virtually free group and an Abelian group is Malcev coherent.
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20M35
Mathematics Subject Classification ID: 68Q45
zbMATH DE Number: 5534967
zbMATH Keywords: knot groups
zbMATH Keywords: Dehn presentations
zbMATH Keywords: complete rewriting systems
zbMATH Keywords: free semigroups
zbMATH Keywords: Malcev congruences
zbMATH Keywords: Malcev coherent semigroups
zbMATH Keywords: Malcev presentations
zbMATH Keywords: polycyclic groups
reviewed by: Yuji Kobayashi
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.10.006
OpenAlex ID: W2005179738
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DOI: 10.1016/J.JPAA.2008.10.006
description: scientific article; zbMATH DE number 5534967