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

TL;DR

This paper investigates when finite distributive lattices can be represented by principal congruences of finite lattices, establishing a necessary and sufficient condition for those with a join-irreducible unit element.

Contribution

It provides a characterization of principal congruence representability for finite distributive lattices with a join-irreducible unit, including a necessary and sufficient condition.

Findings

Identifies a necessary condition for representability.

Proves the condition is sufficient for lattices with a join-irreducible unit.

Advances understanding of lattice congruence structures.

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.