Basic questions for general algebras
type: Item
instance of: scholarly article
title: Basic questions for general algebras
zbMATH Open document ID: 0545.08008
DOI: 10.1007/BF01191487
author: Peter Perkins
published in: Algebra Universalis
publication date: 1984-01-01T00:00:00Z
review text: This paper gives a simple method of generating inherently non-finitely based finite algebras, that is, finite algebras with the property that any locally finite variety which contains such an algebra does not have a finite basis for its laws. The algebras in question are groupoids so the laws are of the form \(t=u\) where the terms t and u are monomials. With each such term can be associated a binary tree with labelled leaves. A set P of terms is said to be closed under subtree free labelling if it contains every term obtained from one of its terms by taking a subtree and labelling the leaves arbitrarily with distinct variables. A term is basic if every subterm is a variable or of the form (v.s) or (s.v) where v is a variable. It is linear if no variable occurs more than once. The main result is: Theorem: Let A be a groupoid which generates a locally finite variety. Let P be a set of linear basic terms with unbounded lengths which is closed under subtree free labelling. If \(t\in P\) and \(t=u\) a law of A imply \(u\in P\) and \(\{xy=yx\}\vdash t=u\) then A is not contained in any finitely based, locally finite variety.- Some examples of such sets of terms and small groupoids with the required properties are given.
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 3867405
zbMATH Keywords: non-finitely based finite algebras
zbMATH Keywords: locally finite variety
zbMATH Keywords: groupoids
zbMATH Keywords: binary tree with labelled leaves
zbMATH Keywords: set of linear basic terms
reviewed by: Sheila Oates-Williams
MaRDI profile type: Publication
cites work: A CLOSED SET OF LAWS WHICH IS NOT GENERATED BY A FINITE SET OF LAWS
cites work: Finite equational bases for finite algebras in a congruence-distributive equational class
cites work: Identities in Finite Algebras
cites work: A proof of Baker's finite-base theorem on equational classes generated by finite elements of congruence distributive varieties
cites work: Equational Bases for Lattice Theories.
cites work: Equational Bases and Nonmodular Lattice Varieties
cites work: Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties
cites work: A new product of algebras and a type reduction theorem
cites work: Unsolvable problems for equational theories
cites work: Bases for equational theories of semi-groups
cites work: Finite groupoids without finite bases for their identities
cites work: Baker's finite basis theorem
full work available at URL: https://doi.org/10.1007/bf01191487
OpenAlex ID: W1990386459
description: scientific article; zbMATH DE number 3867405