Is there a way to find all or some functions which "aggregate" numbers and are non-isomorphic to addition. I mean functions which are commutative and associative:
$f(x,y)=f(y,x)$
$f(x,f(y,z))=f(f(x,y),z)$
Do you know examples?
EDIT: So of I want to exclude trivial solutions which are isomorphic to addition: $f(x,y)=g(h(x)+h(y))$
Multiplication. Perhaps more interesting, $$f(x,y)=x^{\log y}$$ which is defined for positive $x$ and $y$. The trick is to note that this is $e^{\log x\log y}$ and this makes it easy to prove the properties. Another example is $$f(x,y)=\root3\of{x^3+y^3}$$
EDIT: For an example which is "not isomorphic to addition," I think $$f(x,y)=\max(x,y)$$ will do.