An extension theorem for planar semimodular lattices

TL;DR

This paper proves that every finite distributive lattice can be represented as the congruence lattice of a specially constructed rectangular lattice where all congruences are principal, extending the theoretical understanding of lattice representations.

Contribution

It introduces an extension theorem showing that any finite distributive lattice can be realized as a congruence lattice of a rectangular lattice with all principal congruences.

Findings

Every finite distributive lattice can be represented as a congruence lattice of a rectangular lattice.

The constructed lattice ensures all congruences are principal.

The result strengthens the connection between distributive lattices and rectangular lattices.

Abstract

We prove that every finite distributive lattice $D$ can be represented as the congruence lattice of a rectangular lattice $K$ in which all congruences are principal. We verify this result in a stronger form as an extension theorem.