On slim rectangular lattices

2022/05/23 by Grätzer, George
#06 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2205.10966

Abstract

Let L be a slim, planar, semimodular lattice (slim means that it does not contain an \mathsf M₃-sublattice). We call the interval I = [o, i] of L rectangular, if there are complementary a, b ∈ I such that a is to the left of b. We claim that a rectangular interval of a slim rectangular lattice is also a slim rectangular lattice. We will present some applications, including a recent result of G. Czédli. In a paper with E. Knapp about a dozen years ago, we introduced natural diagrams for slim rectangular lattices. Five years later, G. Czédli introduced \E C₁-diagrams We prove that they are the same.

Related

• Another research note
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• Notes on planar semimodular lattices. IX. C₁-diagrams
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• Diagrams and rectangular extensions of planar semimodular lattices