Minimal representations of a finite distributive lattice by principal congruences of a lattice

TL;DR

This paper characterizes finite distributive lattices that can be minimally represented as principal congruences of a finite lattice, focusing on those with at most two dual atoms, thus clarifying the structure of such representations.

Contribution

It provides a characterization of finite distributive lattices with minimal principal congruence representations, specifically when the lattice has at most two dual atoms.

Findings

Finite distributive lattices with at most two dual atoms can have minimal principal congruence representations.

The paper identifies conditions under which such minimal representations exist.

It advances understanding of the structure of congruence lattices in finite lattices.

Abstract

Let the finite distributive lattice $D$ be isomorphic to the congruence lattice of a finite lattice $L$. Let $Q$ denote those elements of $D$ that correspond to principal congruences under this isomorphism. Then $Q$ contains $0,1 \in D$ and all the join-irreducible elements of $D$. If $Q$ contains exactly these elements, we say that $L$ is a minimal representations of $D$ by principal congruences of the lattice $L$. We characterize finite distributive lattices $D$ with a minimal representation by principal congruences with the property that $D$ has at most two dual atoms.