A sufficient condition for the non-finite basis property of semigroups.
type: Item
instance of: scholarly article
title: A sufficient condition for the non-finite basis property of semigroups.
zbMATH Open document ID: 1292.20065
author: Edmond W. H. Lee
published in: Monatshefte für Mathematik
publication date: 2012-11-29T00:00:00Z
review text: The author proves that a semigroup \(S\) has no finite identity basis whenever there is an integer \(n\geq 2\) such that (i) \(S\) does not satisfy the identity \(x^ny^nx^n=x^n(y^nx^n)^{n+1}\); (ii) \(S\) satisfies the identities \(xy_1^ny_2^n\cdots y_k^nx=xy_k^ny_{k-1}^n\cdots y_1^nx\) for \(k=2,3,\ldots\) and the identities \(x^{n+2}=x^2\), \(x^{n+1}yx=xyx=xyx^{n+1}\), \((x^ny^n)^n=(y^nx^n)^n\). The last identity in (ii) can be omitted provided that for any distinct \(x,y\in S\), there exist idempotents \(e,f\in S\) such that \(ex\neq ey\) and \(xf\neq yf\). As an application, the author gives an alternative proof of the result by \textit{W. T. Zhang} and \textit{Y. F. Luo} [Bull. Aust. Math. Soc. 84, No. 3, 484-491 (2011; Zbl 1243.20072)] that the identities of the 6-element \(J\)-trivial semigroup \(L=\langle d,e\mid d^2=d,\;e^2=e,\;ded=0\rangle\) have no finite basis. Moreover, if \(T\) is any group of finite exponent or any commutative semigroup of index at most 2, then the product \(L\times T\) is not finitely based. The author also shows that the variety generated by \(L\) has a largest proper subvariety and this subvariety is not finitely based as well.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 6111284
zbMATH Keywords: semigroups
zbMATH Keywords: semigroup identities
zbMATH Keywords: semigroup varieties
zbMATH Keywords: finite basis property
zbMATH Keywords: finite basis problem
zbMATH Keywords: non-finitely based semigroups
zbMATH Keywords: bases of identities
MaRDI profile type: Publication
full work available at URL: s00605-011-0341-1
OpenAlex ID: W2023165224
cites work: Q4848740
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DOI: 10.1007/S00605-011-0341-1
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description: scientific article; zbMATH DE number 6111284