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

2021/03/07 by Gábor Czédli, Czédli, Gábor, George Grätzer +1 · 2 citations
Computer Science · #06C10 #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rough Sets and Fuzzy Logic #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2103.04458

openalex publication_date 2021/03/07 · openalex created_date 2021/03/15 · openalex updated_date 2026/07/28

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 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 \it Three-pendant Three-crown Property. The proof is based on the first author's papers: 2014 (multifork extensions), 2017 (\mathcal C₁-diagrams), and a recent paper (lamps), introducing the tools we need.

Cited by

• Slim patch lattices as absolute retracts and maximal lattices
• Using the Swing Lemma and C₁-diagrams for congruences of planar semimodular lattices

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