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.