3 - 3 lattice inclusions imply congruence modularity

The complexity of a lattice polynomial is defined inductively, with variables having complexity 0. If p=plv'"vOk or p=plA'''/XOk is the canonical expression of the polynomial O, then the complexity c(p) = l+max{c(p~):l<-i-k}. An n-k lattice inclusion is an inclusion of the form p <-o-with c(p)<-n and c(o-)-k. In this note we use the main result of Day [1] to show that if all the congruence lattices of algebras in a variety satisfy a fixed, nontrivial 3-3 lattice inclusion, then they are all modular. Let q be an element of a distributive lattice D. Then D[q] will denote the lattice obtained by "doubling" the element q, i.e., D[q] = (D-{q}) U {(q, 0), (q, 1)} ordered bya<bifa, beD-{q}anda<binD, a<(q, 0) if a < q in D, (q, 1)<b if q ~-b in D, and (q, 0) < (q, 1). Day showed that if a lattice identity a fails in D[q] for some distributive lattice D and some q ~D, then any congruence variety satisfying e is modular. We prove our result by constructing for each nontrivial 3-3 lattice inclusion)t-< p a distributive lattice D and an element q ~ D such that D[q] fails to satisfy h <-p. LEMMA. In FL(X), let 7rq (i~I, jeJ~) be a meet of variables, and let O'kl (k ~ K, l ~ Lk) be a join of variables. Then (*) A V ~j ~-V A ~ I J~ K Lk fails in FL(X) if and only if (Vi)(:l])(Vk)(3l) var (rrij)N var (r = 0 and (Vk)(al)(Vi)(aj) var (Trii) N var (r = 0.