What is a commutative, associative function which is not isomorphic to addition?

Question

I hope I got the wording for the title right, but I basically mean a smooth function on real numbers or matrices such that $f(x,y)=f(y,x)$ and $f(x,f(y,z))=f(f(x,y),z)$ where the function is not of the form $f(x,y)=g(h(x)+h(y))$.

Does such a function exist?

(But not something like $\max(x,y)$ which is like the limiting function of $(x^N+y^N)^\frac{1}{N}$)

Are $g$ and $h$ supposed to be bijections? If yes, a constant function would do the job. — Arnaud D., Aug 23 '18 at 07:22
@ArnaudD. How do you mean that a constant function is bijective, and wouldn't a constant function be on the form $f(x,y) = g(h(x)+h(y))$? — md2perpe, Aug 23 '18 at 10:32
@md2perpe What I meant was that if $g$ and $h$ are required to be bijective, you can take $f$ to be constant, and it will surely never be equal to $g(h(x)+h(y))$. But of course if $g$ can also be constant then we could have $f(x,y)=g(h(x)+h(y))$. — Arnaud D., Aug 23 '18 at 12:02

Servaes · Answer 1 · 2018-08-23T13:35:20.423

2

How about $f(x,y)=\max\{x,y\}$? Clearly $\max\{x,y\}=\max\{y,x\}$ and $$\max\{x,\max\{y,z\}\}=\max\{\max\{x,y\},z\},$$ and no $g$ and $h$ as you describe exist. It isn't smooth along the diagonal, however.

edited Aug 23 '18 at 13:35

answered Aug 23 '18 at 13:13

Servaes

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It is kind of the limit of a very high order mean. I'm not sure if I covered all mathematical details, but something less "trivial" would be interesting. I'll update the question. – Gere Aug 23 '18 at 14:04

What is a commutative, associative function which is not isomorphic to addition?

1 Answers1