Equations in the theory of Q-distributive lattices

Abstract

A Q-distributive lattice is an algebra (L, ∨, ∧, ▿, 0, 1) of type (2, 2, 1, 0, 0) such that (L, ∨, ∧, 0, 1) is a bounded distributive lattice and ▿ satisfies the equations ▿0 = 0, x ∧ ▿x = x, ▿(x ∨ y) = ▿x ∨ ▿y and ▿(x ∧ ▿y) = ▿x ∧ ▿y. The aim of this paper is to find, for each proper subvariety of the variety of Q-distributive lattices, an equation which determines it, relatively to the whole variety, as well as to give a characterization of the minimum number of variables needed in such equation.