The ascending varietal chain of a variety of semigroups
type: Item
instance of: scholarly article
title: The ascending varietal chain of a variety of semigroups
zbMATH Open document ID: 0614.20042
DOI: 10.1007/BF02573099
author: Yanyan Li
published in: Semigroup Forum
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134865
review text: Let V be a variety of algebras. For every cardinal number \(r<\omega\) let \(V_ r\) be the variety generated by the V-free algebra on r generators. The sequence \(V_ 1\subseteq V_ 2\subseteq...\subseteq V_ r\subseteq...\subseteq V_{\omega}=V\) is called the ascending varietal chain of V. \textit{B. Jónsson}, \textit{G. McNulty} and \textit{R. Quackenbush} [Can. J. Math. 27, 25-31 (1975; Zbl 0305.08003)] showed that for any set S of positive integers there is a variety V such that \(V_ n=V_{n+1}\) if and only if \(n\in S.\) In this paper it is shown that for any set S of integers greater than 1 there is a variety V of semigroups such that \(V_ n=V_{n+1}\) if and only if \(n\in S\). This is best possible in the sense that equality of \(V_ 1\) and \(V_ 2\) implies commutativity of V (and there are only countably many such varieties).
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 3993870
zbMATH Keywords: basic rank
zbMATH Keywords: variety of semigroups
zbMATH Keywords: ascending varietal chain
reviewed by: Peter R. Jones
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3993870