Minimal noncommutative varieties and power varieties
type: Item
instance of: scholarly article
title: Minimal noncommutative varieties and power varieties
zbMATH Open document ID: 0544.20050
author: S. W. Margolis
author: Jean-Éric Pin
published in: Pacific Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: A variety of finite monoids is a class of finite monoids closed under taking submonoids, quotients and \textit{finite} direct products. A language L is a subset of a finitely generated free monoid. The variety theorem of Eilenberg sets up a one to one correspondence between varieties of finite monoids and classes of languages called, appropriately, varieties of languages. If V is a variety of monoids, let PV denote the variety generated by the power set monoids PM, \(M\in V\). The authors show that for any variety, one has \(P^ 3V=P^ 4V\). In fact, if V contains a noncommutative monoid, then \(P^ 3V\) is the variety of all finite monoids; and if V is commutative, then \(P^ 2V=P^ 3V\), a result obtained already by Perrot and Straubing. The proof of the above theorem uses the following result, which the authors also prove: if V is a minimal noncommutative variety, then V is generated either by a noncommutative metabelian group or by the syntactic monoid of one of the languages \(\{a,b\}^*a\), \(a\{a,b\}^*\) or \(\{\) a\(b\}\).
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M35
zbMATH DE Number: 3865588
zbMATH Keywords: variety of finite monoids
zbMATH Keywords: finitely generated free monoid
zbMATH Keywords: variety theorem of Eilenberg
zbMATH Keywords: varieties of languages
zbMATH Keywords: minimal noncommutative variety
reviewed by: Christophe Reutenauer
MaRDI profile type: Publication
full work available at URL: pjm.1984.111.125
OpenAlex ID: W2043288907
DOI: 10.2140/PJM.1984.111.125
description: scientific article; zbMATH DE number 3865588