Hereditarily finitely based monoids of extensive transformations.
type: Item
instance of: scholarly article
title: Hereditarily finitely based monoids of extensive transformations.
zbMATH Open document ID: 1184.20048
author: Edmond W. H. Lee
published in: Algebra Universalis
publication date: 2009-11-25T00:00:00Z
review text: Let \(E_n\) denote the monoid of `extensive' transformations of the ordered set \(\{1,2,\dots,n\}\), that is, the transformations \(\alpha\) with the property that \(i\alpha\geq i\) for all \(i\). The author proves that \(E_n\) is hereditarily finitely based if and only if \(n\leq 3\). By far the most difficult part of the proof is to show that \(E_3\) is hereditarily finitely based. The author notes that \(E_n\) is itself not finitely based, for \(n>4\), by a recent result of \textit{I. A. Goldberg} [in Semigroups and formal languages. Proceedings of the international conference in honour of the 65th birthday of Donald B. McAlister, Lisboa, Portugal, 2005. Hackensack, NJ: World Scientific. 101-110 (2007; Zbl 1131.20043)]. Whether \(E_4\) is finitely based was undetermined at the time the paper was written.
reviewed by: Peter R. Jones
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 03C05
zbMATH DE Number: 5638420
zbMATH Keywords: hereditarily finitely based monoids
zbMATH Keywords: finite basis problem
zbMATH Keywords: partial transformations
zbMATH Keywords: extensive transformations
zbMATH Keywords: order preserving transformations
zbMATH Keywords: finite basis property
zbMATH Keywords: bases of identities
MaRDI profile type: Publication
full work available at URL: s00012-009-0001-7
OpenAlex ID: W1977695021
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DOI: 10.1007/S00012-009-0001-7
description: scientific article; zbMATH DE number 5638420