Hereditarily finitely based semigroups of triangular matrices over finite fields.
type: Item
instance of: scholarly article
title: Hereditarily finitely based semigroups of triangular matrices over finite fields.
zbMATH Open document ID: 1282.20063
author: Wen Ting Zhang
author: Jian Rong Li
author: Yanfeng Luo
published in: Semigroup Forum
publication date: 2013-06-18T00:00:00Z
review text: A semigroup \(S\) is finitely based if there exists some finite set of its identities from which all other identities of \(S\) can be deduced. For any semigroup \(S\), let \(\text{var\,}S\) denote the variety generated by \(S\). A finitely based semigroup \(S\) satisfies the stronger property of being hereditarily finitely based if all semigroups in the variety \(\text{var\,}S\) are finitely based. Let \(\mathcal T_n(F_q)\) be the semigroup of all \(n\times n\) upper triangular matrices over the finite field \(F_q\) of order \(q\). The authors investigate the problem of hereditarily finitely basedness of semigroups \(\mathcal T_n(F_q)\) for any \(n\) and \(q\) (recall, that \(q=p^k\) for some prime number \(p\) and natural number \(k\)). They prove that the variety generated by \(\mathcal T_n(F_2)\) is hereditarily finitely based if and only if \(n\leq 2\). Moreover, the semigroup \(\mathcal T_n(F_q)\) is not hereditarily finitely based for any \(n\geq 3\) and any \(q\).
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M20
zbMATH DE Number: 6176655
zbMATH Keywords: varieties of semigroups
zbMATH Keywords: hereditarily finitely based semigroups
zbMATH Keywords: semigroups of upper triangular matrices
zbMATH Keywords: finite basis problem
zbMATH Keywords: finite semigroups
reviewed by: Leonid M. Martynov
MaRDI profile type: Publication
full work available at URL: s00233-013-9476-9
OpenAlex ID: W1996103656
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DOI: 10.1007/S00233-013-9476-9
description: scientific article; zbMATH DE number 6176655