On atoms in the lattice of quasivarieties (Q1101140)

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?''.