Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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:



Do you know examples?

EDIT: So of I want to exclude trivial solutions which are isomorphic to addition: $f(x,y)=g(h(x)+h(y))$

share|cite|improve this question

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.

share|cite|improve this answer
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.

share|cite|improve this answer
Could you explain? – Trevor Wilson Jan 24 '13 at 0:53
@TrevorWilson ('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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.