A rectangular interval of a rectangular lattice is a rectangular lattice

TL;DR

This paper proves that rectangular intervals within rectangular lattices are themselves rectangular lattices, extending the understanding of lattice substructures and confirming a recent result by G. Czédli.

Contribution

It establishes that rectangular intervals in rectangular lattices inherit the rectangular lattice structure, providing a new insight into lattice substructure properties.

Findings

Rectangular intervals are themselves rectangular lattices.

The result applies to slim, planar, semimodular lattices.

Confirms a recent result by G. Czédli.

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 $o = u_l \wedge u_r$ 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. Cz\'edli.