A generalization of the first Malcev theorem on nilpotent semigroups and nilpotency of the wreath product of semigroups.
type: Item
instance of: scholarly article
title: A generalization of the first Malcev theorem on nilpotent semigroups and nilpotency of the wreath product of semigroups.
zbMATH Open document ID: 1273.20061
author: A. V. Tishchenko
published in: Journal of Mathematical Sciences (New York)
publication date: 2013-04-23T00:00:00Z
review text: A semigroup is `nilpotent in the sense of Malcev, of step \(n\)' (more usually, in English, `of nilpotency class \(n\)') if it satisfies the identity \(U_n=V_n\), where the words \(U_n,V_n\) are defined recursively as follows: \(U_0=x\), \(V_0=y\), \dots, \(U_{k+1}=U_kz_{k+1}V_k\), \(V_{k+1}=V_kz_{k+1}U_k\). The Malcev Theorem in the title asserts that for groups this identity defines nilpotency of class \(n\) in the classical sense. The author first expands this theorem to describe the simple and 0-simple semigroups with this property (they are either nilpotent groups or Brandt semigroups over nilpotent groups, respectively). Most of the paper is devoted to the study of the preservation of nilpotency under the (extended standard) wreath product. Here is a sample theorem (Theorem 4.1). Suppose \(S\) is nilpotent in the usual semigroup sense. If \(R\) is nilpotent in the sense of Malcev, then their wreath product \(S\text{\,wr}_1R\) is nilpotent in the sense of Malcev (and, necessarily, the converse holds). If \(S\) is not nilpotent and \(S\text{\,wr}_1R\) is nilpotent in the sense of Malcev, then \(R\) is necessarily a finite nilpotent group and \(S\) is a uniformly periodic semigroup that is nilpotent in the sense of Malcev. The author goes on to find necessary and sufficient conditions that the wreath product be nilpotent in the sense of Malcev, under the hypothesis that each of \(R\) and \(S\) generates a variety with the property that all of its nilsemigroups are nilpotent.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 6156634
zbMATH Keywords: nilpotent semigroups
zbMATH Keywords: nilsemigroups
zbMATH Keywords: nilpotent in Malcev sense
zbMATH Keywords: wreath products
zbMATH Keywords: varieties
zbMATH Keywords: nilpotency classes
reviewed by: Peter R. Jones
MaRDI profile type: Publication
full work available at URL: s10958-012-1013-y
OpenAlex ID: W2038134510
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DOI: 10.1007/S10958-012-1013-Y
description: scientific article; zbMATH DE number 6156634