Small inherently nonfinitely based finite semigroups
type: Item
instance of: scholarly article
title: Small inherently nonfinitely based finite semigroups
zbMATH Open document ID: 1004.20042
author: Marcel Jackson
published in: Semigroup Forum
publication date: 2002-05-30T00:00:00Z
review text: An algebra \(A\) (or a variety \(\mathcal V\)) is said to be inherently nonfinitely based if it is locally finite, not finitely based and has the property that every locally finite variety \({\mathcal V}'\) for which \(A\in{\mathcal V}'\) (or \({\mathcal V}\subseteq{\mathcal V}'\), respectively) is also not finitely based. It is shown how to construct all ``forbidden divisors'' for the pseudovariety of not inherently nonfinitely based finite semigroups. Several other results concerning finite semigroups that generate an inherently nonfinitely based variety that is minimal amongst those generated by finite semigroups are obtained along the way. For example, aside the variety generated by the well known six element Brandt monoid \(B_2^1\), a variety of this type is necessarily generated by a semigroup with at least 56 elements (all such semigroups with 56 elements are described in the main result).
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 1747983
zbMATH Keywords: finite semigroups
zbMATH Keywords: locally finite varieties of semigroups
zbMATH Keywords: inherently nonfinitely based varieties of semigroups
zbMATH Keywords: pseudovarieties
zbMATH Keywords: forbidden divisors
reviewed by: Leonid M. Martynov
MaRDI profile type: Publication
full work available at URL: s002330010085
OpenAlex ID: W2020451328
DOI: 10.1007/S002330010085
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description: scientific article; zbMATH DE number 1747983