On piecewise-linear homeomorphisms between distributive and anti-blocking polyhedra

TL;DR

This paper generalizes Stanley's piecewise-linear homeomorphism between order and chain polytopes to a broader class of distributive and anti-blocking polyhedra using infinite walks in marked networks.

Contribution

It extends the known PL-homeomorphism from specific polytopes to a large class of distributive polyhedra through a novel approach involving infinite walks in marked networks.

Findings

Generalization of Stanley's PL-homeomorphism to broader polyhedra

Introduction of infinite walks in marked networks as a key tool

Establishment of a large class of distributive polyhedra related by PL-homeomorphisms

Abstract

Stanley (1986) introduced the order polytope and chain polytope of a partially ordered set and showed that they are related by a piecewise-linear homeomorphism. In this paper we view order and chain polytopes as instances of distributive and anti-blocking polytopes, respectively. Both these classes of polytopes are defined in terms of the componentwise partial order on $\mathbb R^n$. We generalize Stanley's PL-homeomorphism to a large class of distributive polyhedra using infinite walks in marked networks.