# Some preliminary results on the set of principal congruences of a finite lattice

## TL;DR

This paper presents initial findings on characterizing subsets of finite distributive lattices that correspond to principal congruences in finite lattices, addressing a problem from a well-known lattice theory text.

## Contribution

It provides preliminary results towards solving the problem of identifying subsets of finite distributive lattices that are isomorphic to principal congruences of finite lattices.

## Key findings

- Partial characterization of such subsets
- Connections between principal congruences and lattice subsets
- Foundational results for future research

## Abstract

In the second edition of the congruence lattice book, Problem 22.1 asks for a characterization of subsets $Q$ of a finite distributive lattice $D$ such that there is a finite lattice $L$ whose congruence lattice is isomorphic to $D$ and under this isomorphism $Q$ corresponds the the principal congruences of $L$. In this note, we prove some preliminary results.