Lamps in slim rectangular planar semimodular lattices

TL;DR

This paper introduces lamps as a new tool to analyze the structure of congruence lattices in slim rectangular lattices, simplifying proofs and revealing new properties.

Contribution

The paper defines lamps in slim rectangular lattices and demonstrates their usefulness in studying congruence lattice properties, including new and existing ones.

Findings

Lamps provide an effective method for analyzing congruence lattices.

The paper proves that congruence lattices satisfy four properties, including two previously known and two new.

Lamps simplify proofs of properties of congruence lattices in slim rectangular lattices.

Abstract

A planar (upper) semimodular lattice $L$ is slim if the five-element nondistributive modular lattice $M_3$ does not occur among its sublattices. (Planar lattices are finite by definition.) Slim rectangular lattices as particular slim planar semimodular lattices were defined by G. Gr\"atzer and E. Knapp in 2007. In 2009, they also proved that the congruence lattices of slim planar semimodular lattices with at least three elements are the same as those of slim rectangular lattices. In order to provide an effective tool for studying these congruence lattices, we introduce the concept of lamps of slim rectangular lattices and prove several of their properties. Lamps and several tools based on them allow us to prove in a new and easy way that the congruence lattices of slim planar semimodular lattices satisfy the two previously known properties. Also, we use lamps to prove that these congruence lattices satisfy four new properties including the two-pendant four-crown property and the forbidden marriage property.