On piecewise-linear homeomorphisms between distributive and anti-blocking polyhedra (Q2221625)

On \({\mathbb R}^n\), let \(\le\) denote the usual componentwise partial order. A polyhedron \(P\subseteq {\mathbb R}^n\) is called distributive if \(x,y\in P\) implies that \(\min\{x,y\},\max\{x,y\}\in P\). It is called anti-blocking if \(y \in P\), \(x\in {\mathbb R}^n_{\ge 0}\) and \(x\le y\) together imply that \(x\in P\). Such polyhedra appear in several combinatorial contexts. With a marked network \(\varGamma\), the authors associate a polyhedron \({\mathscr O}(\varGamma)\) which turns out to be distributive. Under suitable assumptions on the network, they construct a transfer map and give sufficient conditions for it to be bijective and the image of \({\mathscr O}(\varGamma)\) to be an anti-blocking polyhedron. The construction of the transfer map depends on the convergence of series given by infinite walks in directed networks. These transfer maps generalize the \(\mathrm{PL}\)-homeomorphisms between order polytopes and chain polytopes of a finite partially ordered set that were described and utilized by R. P. Stanley in 1986. The paper also discusses duality and concludes with applications and questions. For the entire collection see [Zbl 1456.05001].