Maximal Sublattices of Finite Distributive Lattices. II

Abstract

Let $L$ be a lattice, $J(L) = \{ x \in L|x$ join-irreducible in $L\}$ and $M(L) = \{ x \in L|x$ meet-irreducible in $L\}$. As is well known the sets $J(L)$ and $M(L)$ play a central role in the arithmetic of a lattice $L$ of finite length and particularly, in the case that $L$ is distributive. It is shown that the “quotient set” $Q(L) = \{ b/a|a \in J(L),b \in M(L),a \leqq b\}$ plays a somewhat analogous role in the study of the sublattices of a lattice $L$ of finite length. If $L$ is a finite distributive lattice, its quotient set $Q(L)$) in a natural way determines the lattice of all sublattices of $L$. By examining the connection between $J(K)$ and $J(L)$, where $K$ is a maximal proper sublattice of a finite distributive lattice $L$, the following is proven: every finite distributive lattice of order $n \geqq 3$ which contains a maximal proper sublattice of order $m$ also contains sublattices of orders $n - m,2(n - m)$, and $3(n - m)$; and, every finite distributive lattice $L$ contains a maximal proper sublattice $K$ such that either $|K| = |L| - 1$ or $|K| \geqq 2l(L)$, where $l(L)$ denotes the length of $L$.