Antivarieties of unars
type: Item
instance of: scholarly article
title: Antivarieties of unars
zbMATH Open document ID: 1268.08001
author: A. V. Kartashova
published in: Algebra and Logic
publication date: 2012-12-17T00:00:00Z
review text: A class of algebras is said to be an antivariety if it is definable by anti-identities (an anti-identity is a disjunction of a finite number of negated formulas). In the paper, antivarieties of unars (\(=\) monounary algebras) are dealt with. They form a lattice \({\mathbb L}({\mathcal U})\) with respect to inclusion. It is shown that the unique atom of \({\mathbb L}({\mathcal U})\) is the antivariety defined by \((\forall x)\neg(f(x)=x)\) and that \({\mathbb L}({\mathcal U})\) satisfies neither the ascending nor the descending chain condition. The main result is as follows: There is a continuum of members of \({\mathbb L}({\mathcal U})\) having an independent basis of anti-identities, and, similarly, not having such a basis. Further, a finite unar has an independent basis of anti-identities iff it is definable by the empty set of anti-identities, and iff the unar possesses a one-element cycle.
Mathematics Subject Classification ID: 08A60
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 6116200
zbMATH Keywords: monounary algebra
zbMATH Keywords: antivariety of unars
zbMATH Keywords: lattice of antivarieties of unars
zbMATH Keywords: anti-identity
reviewed by: Danica Jakubiková-Studenovská
MaRDI profile type: Publication
full work available at URL: s10469-011-9147-4
OpenAlex ID: W2087462077
cites work: Q4935937
cites work: Q5834648
cites work: Q4517143
cites work: Lattices of relative colour-families and antivarieties
cites work: Q3866167
cites work: Q5598339
DOI: 10.1007/S10469-011-9147-4
description: scientific article; zbMATH DE number 6116200