Characterizing fully principal congruence representable distributive lattices

Motivated by a recent paper of G. Gr\"atzer, a finite distributive lattice $D$ is said to be 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. Based on earlier results of G. Gr\"atzer, H. Lakser, and the present author, we prove that a finite distributive lattice $D$ is fully principal congruence representable if and only if it is planar and it has at most one join-reducible coatom. Furthermore, even the automorphism group of $L$ can arbitrarily be stipulated in this case. Also, we generalize a recent result of G. Gr\"atzer 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.