Simplex faces of order and chain polytopes

TL;DR

This paper proves that in order and chain polytopes, k-cliques in their 1-skeletons correspond to (k-1)-faces that are simplices, generalizing known descriptions of edges and 2-faces.

Contribution

It establishes a general correspondence between cliques and faces in order and chain polytopes, extending previous specific cases.

Findings

k-cliques correspond to (k-1)-faces that are simplices

Generalizes explicit descriptions of edges and 2-faces

Provides a unified understanding of face structures in these polytopes

Abstract

It will be proved that a $k$-clique in the $1$-skeleton of either the order polytope or the chain polytope corresponds to the $(k-1)$-face, which is a simplex, in each polytope. These results generalize the known explicit descriptions of edges and triangular $2$-faces of each polytope.