  Let $Q$ be a subset of a finite distributive lattice $D$. An algebra $A$
represents the inclusion $Q\subseteq D$ by principal congruences if the
congruence lattice of $A$ is isomorphic to $D$ and the ordered set of principal
congruences of $A$ corresponds to $Q$ under this isomorphism. If there is such
an algebra for every subset $Q$ containing $0$, $1$, and all join-irreducible
elements of $D$, then $D$ is said to be fully (A1)-representable. We prove that
every fully (A1)-representable finite distributive lattice is planar and it has
at most one join-reducible coatom. Conversely, we prove that every finite
planar distributive lattice with at most one join-reducible coatom is fully
chain-representable in the sense of a recent paper of G. Gr\"atzer. Combining
the results of this paper with another paper by the present author, it follows
that every fully (A1)-representable finite distributive lattice is "fully
representable" even by principal congruences of finite lattices. Finally, we
prove that every chain-representable inclusion $Q\subseteq D$ can be
represented by the principal congruences of a finite (and quite small) algebra.
