Commutative associative rational binary operations

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.

• What are $x,y$? – Mark Sapir Aug 12 '13 at 6:08
• @TheMaskedAvenger: If these are real numbers, then apply the third operation of the question to $x=y=0$ or $x=5, y=-5$. – Mark Sapir Aug 12 '13 at 6:44
• James, what do you mean by "nondegenerate operation"? I assume it is a kind of partial function... – Andrés E. Caicedo Aug 12 '13 at 7:00
• I think all that is meant is that formal equations are satisfied: we have a rational function in two variables $f(x, y)$ such that $f(f(x, y), z) = f(x, f(y, z))$ in the field of rational functions on three variables, and $f(x, y) = f(y, x)$. Nondegenerate might include nonconstant. – Todd Trimble Aug 12 '13 at 7:50
• In case it wasn't clear in my earlier comment, "rational function" might be a misleading (although standard) term. The point was to avoid discussion of set-theoretic functions, 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, I think that's what he intends. – Todd Trimble Aug 12 '13 at 12:07

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 $(x+y)/(1-xy)$.
• This is exactly the sort of picture I was hoping for. Is a more explicit description of such functions $R$ available? Also, which such $R$'s are homogeneous? – James Propp Aug 12 '13 at 16:19
• A special case of this answer (which seems to cover the examples in your question) is the theory of Formal Groups. That theory has been very well developed, and in that case the function $R$ is the log function. If you google you'll find tons of survey articles. There's very little machinery needed: just power series – David White Aug 12 '13 at 17:14