TITLE: Are groups in (Var/k, rational maps) necessarily algebraic groups?
QUESTION [8 upvotes]: Let $RVar$ be the category which has complex algebraic verieties as objects and rational maps as morphisms [Edit: for $RVar$ to be a category, the rational maps have to be dominant]. 
Let's consider a group object $G$ in this category, i.e. a tuple $(G,\mu,e,\iota)$ where $\mu:G\times G \rightarrow G$ , $e:* \rightarrow G$, and $\iota: G \rightarrow G$ are rational maps satisfying the usual commutative diagrams defining a group structure.
[Edit: in the light of the comments, e.g. the observation that rational maps have better be dominant, the identity $e:*\rightarrow G$ doesn't seem to make much sense; same for the diagram for the inverse, then; what people, among whom Weil, actually considered were "rational group chunks" in which there's only an associative rational $\mu$, and you can ask the same question(s)].
Just out of curiosity, two natural questions:

Is such a $G$ necessarily an algebraic group?
That is: is it the case that for any $(G,\mu,e,\iota) \in Grp(RVar)$, there exists an algebraic group $(G',\mu',e',\iota')$ and a birational map $\varphi: G\rightarrow G'$ such that "$\varphi$ intertwines the operations of $G$ and $G'$"?
Analogous question in the holomorphic/meromorphic setting.

REPLY [18 votes]: Yes.  This result usually goes under the name of Weil's theorem on group chunks.
The original reference is:
A. Weil, On algebraic groups of transformations, Am. J. Math, 77, 1955, 355 - 381.
In model theory this theorem has taken on a life of its own and is the basis of Hrushovski's theorem on generically presented groups.  A model theoretic treatment of Weil's theorem, including a weakening where one is allowed to use negative powers of the Frobenius, may be found in 
E. Bouscaren, Model theoretic versions of Weil's theorem on pregroups, in The Model Theory of Groups, Notre Dame Mathematical Lectures, Number 11
The analogous results are known at least for compactifiable complex analytic groups and real algebraic groups.  The following paper gives a good general treatment of these problems.
L. van den Dries, Weil's group chunk theorem: a topological setting, Ill. J. Math, 34 no. 1, 1990, 127 - 139.