A rectangular interval of a rectangular lattice is a rectangular lattice

2022/01/20 by Grätzer, G.
#06C10 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2201.08327

Abstract

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 u_l, u_r ∈ [o, i] - \o,i\ such that o = u_l \wedge u_r and i = u_l \vee u_r, where u_l is to the left of u_r. We prove that a rectangular interval of a rectangular lattice is a rectangular lattice. As an application, we get a recent result of G. Czédli.

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