Unstable operations in a generalized cohomology theory $E$ give rise to a functor from the category of algebras over $E_*$ 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 $E_*$ 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.