Q-universal varieties of bounded lattices

A quasivariety K of algebraic systems of finite type is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively. It is known that, for every variety K of (0, I)-lattices, if K contains a finite non-distributive simple (0, l)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, I)-lattices. This paper shows that in general the opposite implication is not true. A family (A(i) : i < 2(w)) of locally finite varieties of (0, I)-lattices is exhibited each of which contains no simple non-distributive (0, I)-lattice and each of which is Q-universal.