Finiteness properties of varieties and the restriction to finite algebras.
type: Item
instance of: scholarly article
title: Finiteness properties of varieties and the restriction to finite algebras.
zbMATH Open document ID: 1073.20052
author: Marcel Jackson
published in: Semigroup Forum
publication date: 2005-06-06T00:00:00Z
review text: Answering a question asked by the reviewer [Sci. Math. Jpn. 53, No. 1, 171-199 (2001; Zbl 0990.20039)], the author exhibits an example of a semigroup variety \(\mathbf V\) generated by a 7-element monoid such that every finite semigroup in \(\mathbf V\) has a finite identity basis while some subvariety of \(\mathbf V\) is not finitely based (Theorem~4.3). He also constructs two finite monoids \(M_1\) and \(M_2\) with the following properties: each \(M_i\) generates a limit monoid variety while in the semigroup variety generated by each \(M_i\) every proper finitely generated subvariety is finitely based but some proper subvariety is not (Proposition~5.1). For general algebras, the author observes (Theorem~7.1) that a combination of results by \textit{R.~McKenzie} [Int. J. Algebra Comput. 6, No. 1, 49-104 (1996; Zbl 0844.08011)] and \textit{R.~Willard} [Trans. Am. Math. Soc. 349, No. 7, 2755-2774 (1997; Zbl 0870.03010)] leads to the undecidability of the following decision problems: given a finite algebra, decide whether or not the variety it generates is hereditarily finitely based or is hereditarily finitely generated or has only finitely many subvarieties.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 03D40
zbMATH DE Number: 2173802
zbMATH Keywords: semigroup varieties
zbMATH Keywords: finite basis property
zbMATH Keywords: limit varieties
zbMATH Keywords: hereditarily finitely based varieties
zbMATH Keywords: hereditarily finitely generated varieties
zbMATH Keywords: small varieties
zbMATH Keywords: finite semigroups
MaRDI profile type: Publication
full work available at URL: s00233-004-0161-x
OpenAlex ID: W1963603096
DOI: 10.1007/S00233-004-0161-X
description: scientific article; zbMATH DE number 2173802