Maximal sublattices of finite distributive lattices. II

Abstract

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