Abelian Sandpile Model Of A Directed Multi-graph

Abstract

In this project, we study the combinatorics, algebraic and algebraic geometry of an Abelian Sandpile Model (ASM). We aim to explore the Algebraic properties of the Abelian Sandpile Model (ASM). We also aim to study the algebraic geometry of the Abelian Sandpile Model (ASM). In this model, we consider a directed multi graph with a sink vertex which is accessible from every other vertex and associates with it a commutative monoidM, a commutative semigroupS and a commutative GroupG which finally turns out to be the sandpile monoid, sandpile semigroup and sandpile group respectively. We observe that the sandpile group is the unique minimal ideal of the sandpile monoid [21]. We study the combinatorial structure of our model and the connections between the algebraic structure of the sandpile monoid, the sandpile semigroup and the sandpile group.