Maximal Sublattices and Frattini Sublattices of Bounded Lattices

We will address these questions in order, and provide good partial answers, especially for nite lattices which are bounded homomorphic images of a free lattice. Recall that a nite lattice is bounded if and only if it can be obtained from the one element lattice by a sequence of applications of Alan Day's doubling construction for intervals. In particular, nite distributive lattices are bounded. On the other hand, we do not have a complete solution for any of the above problems. The main results of this paper can be summarized as follows. (1a) For any k > 0, there exists a nite lattice L which has more than jLj maximal sublattices. (1b) A nite bounded lattice L has at most jLj maximal sublattices. (2a) There exist arbitrarily large nite (or even countably in nite lattices) with a maximal sublattice S such that jSj = 14. (2b) For any " > 0, there exists a nite bounded lattice L with a maximal sublattice S such that jSj < "jLj. (3a) There exist in nitely many lattice varieties V such that every nite nontrivial lattice L 2 V is isomorphic to (L) for some nite lattice L 2 V. (3b) Every nite bounded lattice L can be represented as (K) for some nite bounded lattice K (not necessarily in V(L)).