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

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2 Answers 2

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.

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But these functions are isomorphic to additions. I'll add an explanation to my question... :) –  Gerenuk Jan 23 '12 at 12:54
    
Hmm, max() isn't "exactly" isomorphic to addition, but in a way it is still a limit and therefore not so interesting for me :( Since $\max(x,y)=\lim_{k\to\infty} \sqrt[k]{x^k+y^k}$ –  Gerenuk Jan 24 '12 at 12:53

Looks like Hilbert's 13'th. The answer is no.

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Could you explain? –  Trevor Wilson Jan 24 '13 at 0:53
    
@TrevorWilson (en.wikipedia.org/wiki/Hilbert's_thirteenth_problem) probably explains better. The actual Arnold's proof is very instructive, but I don't have the link in English. –  user58697 Jan 24 '13 at 19:04
    
I read that, but I didn't see its significance for the problem at hand. I think you should add some explanation to your answer. –  Trevor Wilson Jan 24 '13 at 19:20

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