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

Document Type : Research Paper

Authors

1 Bolyai Institute, University of Szeged, Szeged, Aradi H6720 Hungary

2 University of Manitoba, Canada

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 slim if M3 it is not a sublattice of L. In his 2016 monograph, “The
Congruences of a Finite Lattice, A 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 Three-pendant Three-crown Property. The
proof is based on the first author’s papers: 2014 (multifork extensions), 2017
(C1-diagrams), and a recent paper (lamps), introducing the tools we need.

Keywords

References

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Categories and General Algebraic Structures with Applications
Volume 16, Issue 1
January 2022
Pages 1-28

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