Gábor Cz´edli - Bolyai Institute, University of Szeged, Szeged, Aradi H۶۷۲۰ Hungary
George Gr¨atzer - University of Manitoba, Canada

The systematic study of planar semimodular lattices started in۲۰۰۷ 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 M۳ it is not a sublattice of L. In his ۲۰۱۶ 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: ۲۰۱۴ (multifork extensions), ۲۰۱۷(C۱-diagrams), and a recent paper (lamps), introducing the tools we need.

Rectangular lattice, patch lattice, slim semimodular lattice, Congruence lattice, lattice congruence, Three-pendant Three-crown Property