As I understand it, there's basically two approaches to multivalued functions. One of them is that a multivalued function $\mathbb{C} \rightarrow \mathbb{C}$ is just a relation $\mathbb{C} \rightarrow \mathbb{C}$. This works to some extent, e.g. in the Gelfond-Schneider theorem and the explicit solution of the general cubic. However, it seems to be ill-suited for the purposes of calculus, and naive attempts at fixing these kinds of problems tend to fail. To solve this issue, I think there's another approach to multivalued functions via covering spaces. Basically, we declare something along the following lines:
Definition. A multivalued mapping $\mathbb{C} \rightarrow \mathbb{C}$ consists of a set of isolated points $X \subseteq \mathbb{C}$ together with a holomorphic function $U(\mathbb{C} \setminus X) \rightarrow \mathbb{C}$, where $U(J)$ is notation for the universal covering space of $J$.
Something like that. I don't know what such things are actually called, but they generalize meromorphic functions (because we're allowing the singularities to be essential). I'll just call them multivalued mappings here. Anyway, it seems reasonable that such things should form a ring. I've read that the set of meromorphic functions on a connected domain forms a field, so I suppose that the entities under discussion do, too. Seems to make sense. But here's something I really don't understand:
Question. Is there a sensible way of composing these multivalued mappings, making them into a "partial monoid" or something like that? If so, how is this defined, and which pairs of multivalued mappings are composable?
