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: $f(x,y)=g(h(x)+h(y))$