Another research note

TL;DR

This paper explores properties of slim, planar, semimodular lattices, proving that rectangular intervals are themselves rectangular lattices and establishing that natural diagrams coincide with $ ext{C}_1$-diagrams, inheriting their advantageous features.

Contribution

It demonstrates that rectangular intervals in such lattices are rectangular and links natural diagrams to $ ext{C}_1$-diagrams, unifying their properties.

Findings

Rectangular intervals are rectangular lattices.

Natural diagrams are equivalent to $ ext{C}_1$-diagrams.

Natural diagrams inherit properties of $ ext{C}_1$-diagrams.

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. Cz\'edli. In a 2017 paper, G. Cz\'edli 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.