Infinitely many new properties of the congruence lattices of slim semimodular lattices

Abstract

Slim planar semimodular lattices (SPS lattices or slim semimodular lattices for short) were introduced by G. Grätzer and E. Knapp in 2007. More than four dozen papers have been devoted to these (necessarily finite) lattices and their congruence lattices since then. In addition to distributivity, the congruence lattices of SPS lattices satisfy seven known properties. Out of these seven properties, the first two were published by G. Grätzer in 2016 and 2020, the next four by the present author in 2021, and the seventh jointly by G. Grätzer and the present author in 2022. Here we give two infinite families of new properties of the congruence lattices of SPS lattices. These properties are independent. We also present stronger versions of these properties but not all of them are independent, and improve three out of the seven previously known properties. The approach is based on lamps, which we introduced in a 2021 paper. In addition to using the 2021 results, we need to prove some easy new lemmas on lamps.