What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.)


Feel free to re-tag if you can think of something better than "algebra".


Clarification: I intended that $x$ and $y$ denote complex numbers; that the operations be defined almost everywhere; and that the functions not be constant.



Commutative associative rational binary operations
Many examples of such operations can be constructed as follows.


Let $F$ be a meromorphic function which has a rational addition theorem.
This means that $F(u+v)=R(F(u),F(v))$ for all (complex) $u$ and $v$, where $R$ is a
rational function. Then $R$ gives you a rational commutative associative operation.


All such meromorphic functions $F$ have been classified. There is a theorem of
Weierstrass that every such $F$ is either rational, or a rational function of $\exp(au)$,
or an elliptic function. This gives plenty of examples of operations.
For example, taking $F(u)=\tan u$ gives operation $(x+y)/(1-xy)$.


In the case of elliptic functions, you can even obtain 1-parametric families of
such operations. 


