Another research note

Abstract

Let $L$ be a slim, planar, semimodular lattice (slim means that it does not contain ${\mathsf{M}}_3$-sublattices). We call the interval $I=[o,i]$ of $L$ \emph{rectangular}, if there are $u_l,u_r\in [o,i]-\{o,i\}$ such that $i=u_l\vee u_r$ and $o=u_l\wedge u_r$ where $u_l$ is to the left of $u_r$. \emph{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 \emph{${\mathcal{C}}_1$-diagrams}. We revisit the concept of \emph{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. \emph{The second result} shows that for a slim rectangular lattice, a~natural diagram is the same as a ${\mathcal{C}}_1$-diagram. Therefore, natural diagrams have all the nice properties of ${\mathcal{C}}_1$-diagrams.

arXiv commentIncorporated in a longer paper