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.

function" might be a misleading (although standard) term. The point was to avoid discussion of set-theoreticfunctions, and interpret a rational "function" purely formally as an element of the field of fractions of the ring of polynomials. I think thinking of actual functions and their domains is distracting us from what the OP intends, which is really a purely formal problem (and one of potential interest). At least, Ithinkthat's what he intends. – Todd Trimble♦ Aug 12 '13 at 12:07