Finite distributive lattices and doubly irreducible elements

For a finite ordered set G let D(G) denote the family of all distributive lattices L such that G both generates L and is the set of doubly irreducible elements of L. We provide a characterization for membership in D(G), and by means of this characterization define a natural order relation on D(G). We show that this order is a boolean lattice and we describe the maximal and minimal elements in this lattice. The maximal element is familiar: the free distributive lattice freely generated by the ordered set G.