Notes on congruence lattices and lamps of slim semimodular lattices

TL;DR

This paper explores properties of congruence lattices of slim semimodular lattices using lamps, introduces infinitely many new properties, and provides an algorithm to determine representability of distributive lattices as congruence lattices.

Contribution

It introduces new properties of lamps, strengthens a known property, and develops an exponential-time algorithm for lattice representation decision.

Findings

Infinitely many new properties of congruence lattices are presented.

Lamps are effective tools for analyzing slim semimodular lattices.

An exponential-time algorithm for lattice representation is proposed.

Abstract

Since their introduction by G. Gr\"atzer and E. Knapp in 2007, more than four dozen papers have been devoted to finite slim planar semimodular lattices (in short, SPS lattices or slim semimodular lattices) and to some related fields. In addition to distributivity, there have been seven known properties of the congruence lattices of these lattices. The first two properties were proved by G. Gr\"atzer, the next four by the present author, while the seventh was proved jointly by G. Gr\"atzer and the present author. Five out of the seven properties were found and proved by using lamps, which are lattice theoretic tools introduced by the present author in a 2021 paper. Here, using lamps, we present infinitely many new properties. Lamps also allow us to strengthen the seventh previously known property, and they lead to an algorithm of exponential time to decide whether a finite distributive lattice can be represented as the congruence lattice of an SPS lattice. Some new properties of lamps are also given.