QUASIVARIETIES OF DISTRIBUTIVE LATTICES WITH A QUANTIFIER

It is shown that any subvariety V of the variety of bounded distributive lattices with a quantifier, as considered by Cignoli (1991), contains either uncountably or finitely many quasivarieties depending on whether V contains the 4-element bounded Boolean lattice with a simple quantifier. It is also shown that, in the former case, the quasivarieties contained in V form a lattice which fails to satisfy every nontrivial lattice identity while, in the latter case, they form a chain of length less than or equal to 3.