A Simple Solution to the Word Problem for Lattices

Abstract

Whitman [2] solved the word problem for lattices by giving an explicit construction of the free lattice, FL(X), on a given set of generators X.The solution is the following:For x, y ∊ X, and a, b, c, d ∊ FL(X),(W1)(W2)(W3)(W4)where [p, q] = {x; p ≤ x ≤ q}.The purpose of this note is to give a simple nonconstructive proof that the condition (W4) must hold in every projective (hence every free) lattice. Jonsson [1] has shown that in every equational class of lattices (Wl), (W2), and (W3) hold. Therefore the combination of these results gives a complete nonconstructive solution to the word problem for lattices.