Chip-Firing, Antimatroids, and Polyhedra

Starting from the chip-firing game of Björner and Lovász we consider a generalization to vector addition systems that still admit algebraic structures as sandpile group or sandpile monoid. Every such vector addition language yields an antimatroid. We show that conversely every antimatroid can be represented this way. The inclusion order on the feasible sets of an antimatroid is an upper locally distributive lattice. We characterize polyhedra, which carry an upper locally distributive structure and show that they can be modelled by chip-firing games with gains and losses. At the end we point out a connection to a membership problem discussed by Korte and Lovász.