Finite Sublattices of a Free Lattice

Abstract

It is known that every sublattice A of a free lattice satisfies the following conditions:(W) For all a, b, c, d ∈ A, if ab ≤ c + d, then ab ≤ c or ab ≤ d or a ≤ c+dor b ≤ c + d.(SD) For all u, a, b, c ∈ A, if u = a + b = a + c, then u = a + bc.(SD′) For all u, a, b, c ∈ A, if u = ab = ac, then u = a(b + c).In fact, (W) is one of the four conditions used in Whitman (4) to characterize free lattices, and in Jónsson (3) it was shown that (SD) and (SD′) follow from Whitman's canonical representations of elements of a free lattice.This note is concerned with lattices that satisfy one or more of the above conditions, and especially with finite lattices that satisfy all three conditions.