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

The systematic study of planar semimodular lattices started in2007 with a series of papers by G. Grätzer and E. Knapp. These lattices haveconnections with group theory and geometry. A planar semimodular latticeL is slim if M3 it is not a sublattice of L. In his 2016 monograph, “TheCongruences of a Finite Lattice, A Proof-by-Picture Approach”, the secondauthor asked for a characterization of congruence lattices of slim, planar,semimodular lattices. In addition to distributivity, both authors have previouslyfound specific properties of these congruence lattices. In this paper,we present a new property, the Three-pendant Three-crown Property. Theproof is based on the first author’s papers: 2014 (multifork extensions), 2017(C1-diagrams), and a recent paper (lamps), introducing the tools we need.