Characterizing representability by principal congruences for finite distributive lattices with a join-irreducible unit element

Abstract

For a finite distributive lattice $D$, let us call $Q\subseteq D$ \emph{principal congruence representable}, if there is a finite lattice $L$ such that the congruence lattice of $L$ is isomorphic to $D$ and the principal congruences of $L$ correspond to $Q$ under this isomorphism. We find a necessary condition for representability by principal congruences and prove that for finite distributive lattices with a join-irreducible unit element this condition is also sufficient.