# An open problem on congruences of finite lattices

## TL;DR

This paper surveys the ongoing research on the properties and characterization of congruences in SPS lattices, a special class of slim, planar, semimodular lattices, highlighting significant contributions and open problems in the field.

## Contribution

It provides a comprehensive overview of the development and current state of the problem of characterizing congruences in SPS lattices, including major contributions by Gábor Czédli.

## Key findings

- The Two-cover Property for SPS lattices was established.
- Over 50 papers have advanced understanding of congruences in SPS lattices.
- Open problems remain in fully characterizing congruences of SPS lattices.

## Abstract

Let $L$ be a planar semimodular lattice. We call $L$ \emph{slim}, if it has no $\mthree$ sublattice. Let us define an \emph{SPS lattice} as a slim, planar, semimodular lattice $L$. In 2016, I proved a property of congruences of SPS lattices (Two-cover Property) and raised the problem of characterizing them. Since then, more than 50 papers have been published contributing to this problem. In this survey, I provide an overview of this field with major contributions by G\'abor Cz\'edli.