A rectangular interval of a rectangular lattice is a rectangular lattice

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. Czédli.

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