It has been known for sometime that the Abelian Sandpile is an excellent model of Self Organized Criticality. It has been extensively studied from many points of view. In particular, from the algebraic point of view, the recurrent elements have the structure of a finite Abelian group, whose structure has been studied in the literature. More generally the set of all stable configurations has the structure of a finite Abelian semigroup. The purpose of this talk is to give a detailed introduction to the structure of this semigroup. We prove that the semigroup is a nilpotent ideal extension of the group of recurrent elements. This is the algebraic equivalent of the fact that every long enough sequence of avalanches leads to a recurrent state. We compute the structure of the nilpotent part of the semigroup in the case of a one dimensional sandpile. This gives detailed information about the transient elements of the model including the length of the longest non-recurrent sequence.