An algorithm for determining Φ(L) in finite distributive lattices

Abstract

Following G. Birkhoff, the Frattini sublattice Φ(L) of a latticeL is defined as the intersection of all its maximal proper sublattices. Let ℒ(FD) be the class of all finite distributive lattices. The main aim in this note is to provide a new but elementary characterization of elements in Φ(L),L∈ℒ(FD), and also an extremely simple algorithm for determining the Frattini sublattice of any finite distributive lattice. By applying this algorithm, it is shown that there is a new way to determine the Frattini subalgebra of a finite Stone algebra.