Formal power series: an algebraic approach to the GapP and #P functions

The algebraic structure of GapP and #P functions is introduced by formalizing them as power series ring and semiring, respectively. It is proved that for every invertible GapP function g, P^g=P¹g/, and for all positive integers i, P^g=P to the gⁱ, power. Applying the Ladner theorem for functions, it is shown that P^s=P^[s] for all S if and only if P=PP, where [S] is the subring generated by S. It also concluded that the class of rational series is contained in FP. Considering the group of all invertible functions, it is proved that all functions in the same coset with respect to FP are Turing equivalent, every Turing degree inside GapP except FP contains infinitely many cosets, and P≠PP if and only if the index of FP is infinite