Revisiting the representation theorem of finite distributive lattices with principal congruences

TL;DR

This paper combines two classical results to construct a planar semimodular lattice representing any finite distributive lattice with all congruences principal, advancing the understanding of lattice representations.

Contribution

It merges previous theorems to construct a planar semimodular lattice with all principal congruences representing any finite distributive lattice.

Findings

Constructed a planar semimodular lattice with all principal congruences.

Unified two classical representation theorems.

Extended the techniques from previous work to achieve this result.

Abstract

A classical result of R.\,P. Dilworth states that every finite distributive lattice $D$ can be represented as the congruence lattice of a finite lattice~$L$. A~sharper form was published in G.~Gr\"atzer and E.\,T. Schmidt in 1962, adding the requirement that all congruences in $L$ be principal. Another variant, published in 1998 by the authors and E.\,T. Schmidt, constructs a planar semimodular lattice $L$. In this paper, we merge these two results: we construct $L$ as a planar semimodular lattice in which all congruences are principal. This paper relies on the techniques developed by the authors and E.\,T. Schmidt in the 1998 paper.