Equational closure operator and forbidden semidistributive lattices
type: Item
instance of: scholarly article
title: Equational closure operator and forbidden semidistributive lattices
zbMATH Open document ID: 0711.08013
DOI: 10.1007/BF00970904
author: Kira Adaricheva
author: V. A. Gorbunov
published in: Siberian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let \({\mathcal S}\) be a quasivariety. A subclass \({\mathcal R}\subset {\mathcal S}\) is called an \({\mathcal S}\)-variety if \({\mathcal R}={\mathcal S}\cap {\mathcal V}\) for some variety \({\mathcal V}\). Denote by \(L_ q({\mathcal S})\) the lattice of all subquasivarieties of \({\mathcal S}\). Let Q be the class of all lattices of quasivarieties. The author found some new property of Q-lattices which implies the known Dziobiak's property. It is shown that the operator on \(L_ q({\mathcal S})\) assigning to each subquasivariety its equational closure in \({\mathcal S}\) can be axiomatized and that every coalgebraic lattice is isomorphic to the lattice of \({\mathcal S}\)-varieties of some quasivariety S. The authors characterize ordered sets whose lattices of convex subsets are Q-lattices.
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 4168754
zbMATH Keywords: quasivariety
zbMATH Keywords: lattices of quasivarieties
zbMATH Keywords: Dziobiak's property
zbMATH Keywords: equational closure
zbMATH Keywords: coalgebraic lattice
zbMATH Keywords: lattices of convex subsets
reviewed by: Ivan Chajda
MaRDI profile type: Publication
cites work: Q4132572
cites work: On atoms in the lattice of quasivarieties
cites work: A property of the lattice of equational theories
cites work: Q3956998
cites work: Finite groups with a self-normalizing subgroup of order 6
cites work: Q3963056
cites work: Q4078085
cites work: The convexity lattice of a poset
description: scientific article; zbMATH DE number 4168754