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.