2022/01/31 by George Grätzer, Grätzer, George
Computer Science · Mathematics · #06C10 #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric and Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2201.13343
openalex publication_date 2022/01/31 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
Let L be a slim, planar, semimodular lattice (slim means that it does not contain \mathsf M3-sublattices). We call the interval I = [o, i] of L rectangular, if there are ul, ur ∈ [o, i] - \o,i\ such that i = ul \vee ur and o = ul \wedge ur where ul is to the left of ur. The first result: a rectangular interval of a rectangular lattice is a rectangular lattice. As an application, we get a recent result of G. Czédli. In a 2017 paper, G. Czédli introduced a very powerful diagram type for slim, planar, semimodular lattices, the \emphC1-diagrams. We revisit the concept of natural diagrams I introduced with E.~Knapp about a dozen years ago. Given a slim rectangular lattice L, we construct its natural diagram in one simple step. The second result shows that for a slim rectangular lattice, a~natural diagram is the same as a C1-diagram. Therefore, natural diagrams have all the nice properties of C1-diagrams.