Characterizing fully principal congruence representable distributive lattices

Motivated by a recent paper of G. Gratzer, a finite distributive lattice D is called fully principal congruence representable if for every subset Q of D containing 0, 1, and the set J(D) of nonzero join-irreducible elements of D, there exists a finite lattice L and an isomorphism from the congruence lattice of L onto D such that Q corresponds to the set of principal congruences of L under this isomorphism. A separate paper of the present author contains a necessary condition of full principal congruence representability: D should be planar with at most one join-reducible coatom. Here we prove that this condition is sufficient. Furthermore, even the automorphism group of L can arbitrarily be stipulated in this case. Also, we generalize a recent result of G. Gratzer on principal congruence representable subsets of a distributive lattice whose top element is join-irreducible by proving that the automorphism group of the lattice we construct can be arbitrary.