Multiple and inverse topplings in the Abelian Sandpile Model

The Abelian Sandpile Model is a cellular automaton whose discrete dynamics reaches an out-of-equilibrium steady state resembling avalanches in piles of sand. The fundamental moves defining the dynamics are encoded by the toppling rules. The transition monoid corresponding to this dynamics in the set of stable configurations is abelian, a property which seems at the basis of our understanding of the model. By including also antitoppling rules, we introduce and investigate a larger monoid, which is not abelian anymore. We prove a number of algebraic properties of this monoid, and describe their practical implications on the emerging structures of the model.