Q -universal varieties of bounded lattices

Abstract

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, 1)-lattices, if K contains a finite nondistributive simple (0, 1)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, 1)-lattices. This paper shows that in general the opposite implication is not true. A family (A i : i < 2 ω ) of locally finite varieties of (0, 1)-lattices is exhibited each of which contains no simple non-distributive (0, 1)-lattice and each of which is Q-universal.

Almost ff-universality Implies Q-universality

“…In [5], Adams and Dziobiak exhibited 2 ℵ0 locally finite Q-universal (0, 1)-lattice varieties all of whose non-singleton members have a prime ideal. Yet no such variety is almost universal as a variety of (0, 1)-lattices or 0-lattices or lattices because in all three cases none of the varieties have almost rigid members with more than two elements.…”

“…We prove Theorem 1. 5 Any almost f f -alg-universal quasivariety contains an A-D family, and hence it is Q-universal.…”