Existence of independent quasi-equational bases
type: Item
instance of: scholarly article
title: Existence of independent quasi-equational bases
zbMATH Open document ID: 1452.08006
published in: Algebra and Logic
publication date: 2020-08-03T00:00:00Z
review text: The paper belongs to a series of papers of the author and her collaborators concerning independent quasi-equational bases for quasivarieties of algebraic structures, and the related problem of complexity of lattices of quasivarieties. Let \(\mathbf{K} \subseteq \mathbf{M}\) be classes of algebraic structures of the same similarity type \(\sigma\). As defined by the author, the class \(\mathbf{K}\) has an \emph{independent quasi-equational basis relative to} \(\mathbf{M}\), if \(\mathbf{K} = \mathbf{M} \cap \mathrm{Mod}(\Phi)\) for a set \(\Phi\) of quasi-identities and \(\mathbf{K} \neq \mathbf{M} \cap \mathrm{Mod}(\Phi \setminus \{\varphi\})\) for each \(\varphi \in \Phi\). A quasivariety \(\mathbf{K}\) has an \emph{independent quasi-equational basis} if \(\mathbf{K}\) has an independent quasi-equational basis relative to the class of all structures of type \(\sigma\). A sufficient condition is given for a quasivariety \(\mathbf{K}\) which ensures that \(\mathbf{K}\) contains continuum many subquasivarieties with no independent quasi-equational basis relative to \(\mathbf{K}\). This result is a weaker version of an earlier result given by A. V. Kravchenko, A. M. Nurakunov and the author. Some interesting consequences and applications of the author's result are provided. In particular, the condition holds for any so-called almost \(\mathit{f \, f}\)-universal quasivariety \(\mathbf{K}\). Also some further open problems are formulated.
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 06B15
zbMATH DE Number: 7226858
zbMATH Keywords: quasivariety
zbMATH Keywords: independent quasi-equational basis
author: M. V. Schwidefsky
reviewed by: Anna Romanowska
MaRDI profile type: Publication
full work available at URL: s10469-020-09570-3
OpenAlex ID: W3009396778
Wikidata QID: Q114691253
cites work: Q5834648
cites work: Q4935937
cites work: Covers in lattices of quasivarieties and independent axiomatizability
cites work: Structure of quasivariety lattices. I: Independent axiomatizability
cites work: Structure of quasivariety lattices. II: Undecidable problems
cites work: Q-Universal Quasivarieties of Algebras
cites work: Almost \textit{ff}-universality implies \(Q\)-universality
cites work: On quasi-equational bases for differential groupoids and unary algebras
cites work: Lattices of subclasses. III
cites work: Q5628181
cites work: Quasivarieties of unary algebras with a finite number of cycles
cites work: Q3606621
cites work: On lattice identities satisfied in subquasivariety lattices of varieties of modular lattices
cites work: Complexity of quasivariety lattices.
cites work: Q3822212
cites work: Q4789441
cites work: Lattices of subclasses. II
cites work: Topologische Reflexionen und Coreflexionen
cites work: Q3208855
cites work: Q3417721
cites work: Universality of Small Lattice Varieties
cites work: On almost universal varieties of modular lattices
cites work: Almost ff-universal and q-universal varieties of modular 0-lattices
cites work: Q3417724
cites work: Universally axiomatizable subclasses of locally finite classes of models
cites work: Q3565123
DOI: 10.1007/S10469-020-09570-3
description: scientific article; zbMATH DE number 7226858