Primitive lattice varieties

Abstract

A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition [Formula: see text]. For example, if every finite subdirectly irreducible lattice in a locally finite variety [Formula: see text] satisfies Whitman’s condition [Formula: see text], then [Formula: see text] is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are [Formula: see text] such varieties. Some lattices that fail [Formula: see text] also generate primitive varieties. But if [Formula: see text] is a [Formula: see text]-failure interval in a finite subdirectly irreducible lattice [Formula: see text], and [Formula: see text] denotes the lattice with [Formula: see text] doubled, then [Formula: see text] is never primitive.