A new property of congruence lattices of slim, planar, semimodular lattices

Abstract

The systematic study of planar semimodular lattices started in 2007 with a series of papers by G. Grätzer and E. Knapp. These lattices have connections with group theory and geometry. A planar semimodular lattice $L$ is {\it slim} if $M_3$ it is not a sublattice of $L$. In his 2016 monograph, "The Congruences of a Finite Lattice, A \emph{Proof-by-Picture Approach}", the second author asked for a characterization of congruence lattices of slim, planar, semimodular lattices. In addition to distributivity, both authors have previously found specific properties of these congruence lattices. In this paper, we present a new property, the {\it Three-pendant Three-crown Property}. The proof is based on the first author's papers: 2014 (multifork extensions), 2017 (${\mathcal{C}}_1$-diagrams), and a recent paper (lamps), introducing the tools we need.

17 pages, 10 figures

Cited by in corpus (2)

• Slim patch lattices as absolute retracts and maximal lattices
• Using the Swing Lemma and ${\mathcal{C}}_1$-diagrams for congruences of planar semimodular lattices