Maximal Sublattices of Finite Distributive Lattices. II

Let be a lattice, <!-- MATH $J(L) = \{ x \in L|x$ --> join-irreducible in and <!-- MATH $M(L) = \{ x \in L|x$ --> meet-irreducible in . As is well known the sets and play a central role in the arithmetic of a lattice of finite length and particularly, in the case that is distributive. It is shown that the ``quotient set'' <!-- MATH $Q(L) = \{ b/a|a \in J(L),b \in M(L),a \leqq b\}$ --> plays a somewhat analogous role in the study of the sublattices of a lattice of finite length. If is a finite distributive lattice, its quotient set ) in a natural way determines the lattice of all sublattices of . By examining the connection between and , where is a maximal proper sublattice of a finite distributive lattice , the following is proven: every finite distributive lattice of order which contains a maximal proper sublattice of order also contains sublattices of orders <!-- MATH $n - m,2(n - m)$ --> , and ; and, every finite distributive lattice contains a maximal proper sublattice such that either <!-- MATH $|K| = |L| - 1$ --> or <!-- MATH $|K| \geqq 2l(L)$ --> , where denotes the length of .