Nonfinitely based pseudovarieties and inherently nonfinitely based varieties
type: Item
instance of: scholarly article
title: Nonfinitely based pseudovarieties and inherently nonfinitely based varieties
zbMATH Open document ID: 0794.08007
DOI: 10.1007/BF02573759
author: Robert Cacioppo
published in: Semigroup Forum
publication date: 1994-01-06T00:00:00Z
full work available at URL: https://eudml.org/doc/135272
review text: An algebra is a set together with a finite number of finitary operations. The pseudovariety generated by an algebra \(A\) is the smallest class containing \(A\) and closed under homomorphisms, subalgebras, and finite products. The pseudovariety generated by \(A\) consists of all finite algebras in the variety generated by \(A\). If \(A\) is finitely based then it generates a finitely based pseudovariety. The general question: if \(A\) is nonfinitely based, does it necessarily generate a nonfinitely based pseudovariety? is open. The author shows that if \(A\) has a property called \(\omega\)-bounded, then it generates a nonfinitely based pseudovariety. In this paper he gives two examples of \(\omega\)-bounded algebras, Perkins' semigroup and Lyndon's groupoid and shows that every nonfinitely based finite algebra is either inherently nonfinitely based or \(\omega\)-bounded. Perkins' semigroup falls in the intersection of these classes while Lyndon's groupoid must be \(\omega\)-bounded since it is not inherently nonfinitely based.
Mathematics Subject Classification ID: 08B99
Mathematics Subject Classification ID: 08A62
zbMATH DE Number: 436441
zbMATH Keywords: nonfinitely based pseudovariety
zbMATH Keywords: \(\omega\)-bounded algebras
zbMATH Keywords: Perkins' semigroup
zbMATH Keywords: Lyndon's groupoid
zbMATH Keywords: nonfinitely based finite algebra
zbMATH Keywords: inherently nonfinitely based
reviewed by: Katarzyna Hałkowska
MaRDI profile type: Publication
cites work: Lyndon's groupoid is not inherently nonfinitely based
cites work: Q3934450
cites work: On finite bases for varieties and pseudovarieties
cites work: Bases for equational theories of semi-groups
cites work: PROBLEMS OF BURNSIDE TYPE AND THE FINITE BASIS PROPERTY IN VARIETIES OF SEMIGROUPS
cites work: Q3807431
description: scientific article; zbMATH DE number 436441