Cubical rectangles and rectangular lattices

TL;DR

This paper introduces cubical rectangles within n-dimensional cubes, characterizes them, and constructs new lattice structures called rectangular lattices, revealing their relation to cubical lattices and face lattices of the n-cube.

Contribution

It defines and explores cubical rectangles, constructs rectangular lattices, and establishes their relationship with existing cubical and face lattices.

Findings

Rectangular lattices are closely related to cubical lattices.

Cubical rectangles include all edges and squares of the n-cube.

New posets and lattices are constructed from cubical rectangles.

Abstract

Cubical rectangles are being defined and explored here over the $n-$dimensional geometric cube $Q_n.$ They form a new class of geometric objects that includes all the edges and all the squares of the $n-$cube. We enumerate and characterize them here in order to construct new posets, transforming into special lattices that will be called rectangular lattices. We show that rectangular lattices are closely related to the class of cubical lattices, that is, the face lattice of the $n-$cube.