On the Strong Purity of the Sublattice-Lattice of a Finite Distributive Lattice
C. C. CHEN, K.M. Koh · Tokyo Journal of Mathematics · 1983
Let $L(FD)$ be the class of finite distributive lattices.In this paper we proceed to study the structure of Sub $(L)$ of $L,$ $L\in L(FD)$ , by employ- ing the notion of the Frattini sublattice of $L$ .Following Birkhoff [1],the Frattini sublattice $\Phi(L)$ of a lattice $L$ is the intersection of all proper maximal sublattices of $L$ .Thus, the element $\Phi(L)$ in the lattice Sub $(L)$ is the meet of all dual atoms in Sub $(L)$ .Denote by Sub*(L) the interval $[\Phi(L), L]$ and by $Sub_{*}(L)$ the interval $[\emptyset, \Phi(L)]$ in Sub $(L)$ .The lattice Sub $(L)$ is said to be pure if $Sub_{*}(L)$ forms a Boolean sublattice of Sub $(L)$ , and doubly pure if, in addition, $Sub_{*}(L)$ also forms a Boolean sublattice of Sub $(L)$ .A pure lattice Sub $(L)$ is said to be strongly pure if every atom in Sub$(L)-Sub_{*}(L)$ is contained in (less than) a unique