On the axiomatic rank of varieties generated by a semigroup or monoid with one defining relation
type: Item
instance of: scholarly article
title: On the axiomatic rank of varieties generated by a semigroup or monoid with one defining relation
zbMATH Open document ID: 0671.20056
DOI: 10.1007/BF02573281
published in: Semigroup Forum
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134971
review text: A variety is said to have finite axiomatic rank if it has a system of defining identities in which the number of variables does not exceed some fixed positive integer. A variety is said to be finitely based if it can be defined by a finite number of identities. The authors' main result is that the variety generated by a finitely-generated semigroup (or monoid) with one defining relation has finite axiomatic rank if and only if it is finitely based. He also solves two related problems: he shows that the variety generated by the bicyclic monoid has infinite axiomatic rank; and he gives an example of a finitely based monoid variety generated by a monoid obtained by the adjunction of an identity element to a semigroup, but where the smaller semigroup itself generates a semigroup variety that is not finitely based.
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 4097666
zbMATH Keywords: varieties of semigroups
zbMATH Keywords: finite axiomatic rank
zbMATH Keywords: finitely based
zbMATH Keywords: finitely- generated semigroup
zbMATH Keywords: finitely based monoid variety
author: Shneerson, L. M.
reviewed by: Graham T. Clarke
MaRDI profile type: Publication
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OpenAlex ID: W2012370060
description: scientific article; zbMATH DE number 4097666