Our work is devoted to abstract study of so-called normed semi-groups, i.e., semi-groups having both rich topological and rich algebraic structure, which arise in some problems of non-commutative geometry. We consider in details the most interesting example, namely, an Abel semi-group $\mathscr N(A)$, where $A$ is a von Neumann algebra. The definition of this semi-group is based on the notion of stable equivalence of normal elements of $W^*$-algebras which generalizes the notion of stable equivalence of projectors.