On atoms in the lattice of quasivarieties

descriptionPublicationkeyboard_double_arrow_right Article 01 Feb 1987 English Publisher:Springer Science and Business Media LLCJournal:Algebra Universalis, volume 24, pages 32-35 (issn: 0002-5240, eissn: 1420-8911,

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Authors: Dziobiak, Wiesław;

doi: 10.1007/bf01188380

On atoms in the lattice of quasivarieties

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Abstract

A Q-lattice is a lattice isomorphic to the subquasivariety lattice of a quasivariety of algebraic systems. Every Q-lattice is join semi- distributive. The converse statement is false since every Q-lattice is atomic and its dual is algebraic. The aim of the present paper is to prove the following theorem: ``The join of a finite set X of atoms in any Q-lattice contains at most \(2^{| X|}-1\) atoms'', which allows us to answer the following question: ``Is every finite join semi- distributive lattice a Q-lattice?''.

Related Organizations

Polish Academy of Learning
Poland

Keywords

finite distributive lattice, atoms, Lattices of varieties, subquasivariety lattice of a quasivariety, finite join semi-distributive lattice, Structure theory of lattices, Quasivarieties, Q-lattice

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