On slim rectangular lattices

Abstract

Let $L$ be a slim, planar, semimodular lattice (slim means that it does not contain an ${\mathsf{M}}_3$-sublattice). We call the interval $I=[o,i]$ of $L$ \emph{rectangular}, if there are complementary $a,b\in I$ 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 ${\E C}_1$-diagrams} We prove that they are the same.

arXiv commentarXiv admin note: substantial text overlap with arXiv:2201.08327, arXiv:2201.13343, arXiv:2104.06539