Operators on generalized power series

Given a ring C and a totally (resp. partially) ordered set of "monomials" M-fraktur sign, Hahn (resp. Higman) defined the set of power series C[[M-fraktur sign]] with well-ordered (resp. Noetherian or well-quasi-ordered) support in M-fraktur sign. This set C[[M-fraktur sign]] can usually be given a lot of additional structure: if C is a field and M-fraktur sign a totally ordered group, then Hahn proved that C[[M-fraktur sign]] is a field. More recently, we have constructed fields of "transseries" of the form C[[M-fraktur sign]] on which we defined natural derivations and compositions. In this paper we develop an operator theory for generalized power series of the above form. We first study linear and multilinear operators. We next isolate a big class of so-called Noetherian operators Φ : C[[M-fraktur sign]] → C[[N-fraktur sign]], which include (when defined) summation, multiplication, differentiation, composition, etc. Our main result is the proof of an implicit function theorem for Noetherian operators. This theorem may be used to explicitly solve very general types of functional equations in generalized power series.