Formal plethories

Unstable operations in a generalized cohomology theory E   give rise to a functor from the category of algebras over _E?$E_{?}$ to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this paper I set up a framework for studying the algebra of such functors, which I call formal plethories, in the case where _E?$E_{?}$ is a Prüfer ring. I show that the “logarithmic” functors of primitives and indecomposables give linear approximations of formal plethories by bimonoids in the 2-monoidal category of bimodules over a ring.