Finite basis problem for semigroups of order five or less: generalization and revisitation.
type: Item
instance of: scholarly article
title: Finite basis problem for semigroups of order five or less: generalization and revisitation.
zbMATH Open document ID: 1283.20063
author: Edmond W. H. Lee
published in: Studia Logica
publication date: 2013-08-19T00:00:00Z
review text: In the abstract of the paper the author writes: ``A system of semigroup identities is hereditarily finitely based if it defines a variety all semigroups of which are finitely based. Two new types of hereditarily finitely based identity systems are presented. Two of these system, together with eight systems, establish the hereditary finite basis property of every semigroup of order five or less with one possible exception''. This possible exception is the monoid \(P_2^1=\langle a,b,1\mid a^2=ab=a,\;b^2a=b^2\rangle\). Two semigroups are `distinct' if they are neither isomorphic nor anti-isomorphic. It is known that there exist 1160 pairwise distinct five-element semigroups among which 156 are monoids [see \textit{R. J. Plemmons}, Math. Algorithms 2, 2-17 (1967); ibid. 3, 23 (1968; Zbl 0247.20089); \textit{A. Distler} and \textit{T. Kelsey}, Ann. Math. Artif. Intell. 56, No. 1, 3-21 (2009; Zbl 1204.20075)]. The author asserts that ``using a computer, it is routinely shown that with the exception of the semigroup \(P_2^1\), each of the other 1159 pairwise distinct semigroups of order five satisfies one of the identity systems (0)-(9) or its dual system and so is hereditarily finitely based''. In 2010 Edmunds et al. have shown that 155 five-element monoids are hereditarily finitely based [see \textit{C. C. Edmunds, E. W. H. Lee} and \textit{K. W. K. Lee}, Order 27, No. 1, 83-100 (2010; Zbl 1197.20050)].
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 6199437
zbMATH Keywords: finite semigroups
zbMATH Keywords: semigroup identities
zbMATH Keywords: finite basis problem
zbMATH Keywords: finitely based semigroups
zbMATH Keywords: minimal non-finitely based semigroups
zbMATH Keywords: hereditarily finitely based semigroups
zbMATH Keywords: semigroup varieties
zbMATH Keywords: monoid varieties
zbMATH Keywords: bases of identities
zbMATH Keywords: hereditary finitely based varieties
describes a project that uses: Smallsemi
MaRDI profile type: Publication
full work available at URL: s11225-012-9369-z
OpenAlex ID: W2073188806
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DOI: 10.1007/S11225-012-9369-Z
description: scientific article; zbMATH DE number 6199437