Almost distributive lattice varieties

Abstract

This paper is concerned with the properties of the lattice A of all lattice varieties (or equational classes of lattices). This lattice is of course dually algebraic, as is the lattice of subvarieties of any variety of algebras, but since lattices are congruence distributive, A is also distributive. Like much of the earlier work, our investigations of A are primarily concerned with the bottom part of the lattice. In fact, we introduce a lattice variety whose members we call almost distributive, and study its subvarieties. These lattices first arose in J6nsson and Rival [11], although no name was given to them there. The subdirectly irreducible almost distributive lattices can be characterized by the facts that they have a unique critical quotient c/a (i.e., every non-trivial congruence relation collapses c/a) and that their quotient lattices by con (a, c) are distributive. Alternatively, using Day's notion of a splitting of a quotient (in this case an element) in a lattice, we see that a subdirectly irreducible lattice L is almost distributive if and only if L is isomorphic to the lattice Did] obtained by splitting an element d in a distributive lattice D. In Rose [14], a necessary and sufficient condition was given for D[d], with D a finite distributive lattice, to be subdirectly irreducible. We give here another characterization (Theorem 5.1), which turns out to be more useful for our purpose. Moreover, we show that subdirectly irreducible almost distributive lattices are described by matrices of zeros and ones, and vice versa, and that all subdirectly irreducible almost distributive lattices are completely determined (Theorems 4.10, 4.13 and 5.7). Using these results we can easily investigate most of the properties of the lattice A of all almost distributive lattice varieties (Section 6).