Abstract
The Chip Firing Game (CFG) is a discrete dynamical model used in physics, computer science and economics. It is known that the set of configurationsreachable from an initial configuration (this set is called the \textitconfiguration space) can be ordered as a lattice. We first present a structural result about this model, which allows us to introduce some useful tools for describing those lattices. Then we establish that the class of lattices that are the configuration space of a CFG is strictly between the class of distributive lattices and the class of upper locally distributive (or ULD) lattices. Finally we propose an extension of the model, the \textitcoloured Chip Firing Game, which generates exactly the class of ULD lattices.
Highlights
The Chip Firing Game (CFG) was introduced by Bjorner, Lovasz and Shor in [BLS91] and [BL92]
J is partitioned into M sets mjJjm . ¢ we give Birkhoff’s representation theorem for distributive lattices: J, there is Theorem 2.5 (Birkhoff) [Bir33] A lattice is distributive if and only if it is isomorphic to the lattice of the ideals of the order induced on its meet-irreductibles
The proof is given in two steps: first we show in Theorem 5.2 that it is included in the class of ULD lattices, we show in Theorem 5.6 that it contains it
Summary
The Chip Firing Game (CFG) was introduced by Bjorner, Lovasz and Shor in [BLS91] and [BL92]. The CFG is a discrete dynamical model, with the following evolution rule, called the firing rule: if, when the game is in a configuration σ, a vertex v contains at least as many chips as its outgoing degree, one can transfer a chip from v along each of its outgoing edges to the corresponding vertex. We call this applying the rule v (and we will speak equivalently of v as a vertex or as a rule).
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