On slim rectangular lattices

TL;DR

This paper studies the structure of rectangular intervals within slim, planar, semimodular lattices, showing they preserve slim rectangular properties and exploring diagram equivalences, with applications to recent lattice theory results.

Contribution

It proves that rectangular intervals in slim rectangular lattices are themselves slim rectangular lattices and establishes the equivalence of two diagram types, advancing lattice structural understanding.

Findings

Rectangular intervals inherit slim rectangular lattice structure.

Two diagram types, introduced separately, are proven equivalent.

Applications include recent results in lattice theory.

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. Cz\'edli. In a paper with E. Knapp about a dozen years ago, we introduced natural diagrams} for slim rectangular lattices. Five years later, G. Cz\'edli introduced ${\E C}_1$-diagrams} We prove that they are the same.