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There was a question whether associative, but non-commutative binary operation over the real numbers exist. A trivial answer is the binary operation $x\circ y = x$ or $x\circ y = y$.

As a followup, there was another question, which specifically excluded trivial binary operations of the type $x\circ y = f(x)$ and $x\circ y = g(y)$ and also requiring it to be continuous (almost) everywhere. One answer was $x\circ y = \vert x \vert y$. None of the answers (so far) provided an analytic function.

So, the question is whether there is a binary operation $\circ:S \to \mathbb R$ (where $S \subseteq \mathbb R\times \mathbb R$) which is:

  1. associative: $(x \circ y) \circ z = x \circ (y \circ z)$
  2. non-commutative: $\exists (x,y)$ such that $x \circ y \ne y \circ x$
  3. non-trivial: $\nexists f(x)$ such that $x\circ y = f(x)$ or $x\circ y= f(y)$
  4. analytic: $$x \circ y = \sum_{m,n=0}^\infty a_{mn}(x-x_0)^m (y-y_0)^n$$
Danijel
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  • Restricting $S$ to ${ -1, 1}^2$ suffices, but isn't satisfying. This way $x^2 = |x|,$ and so the analytic function $x \circ y = x^2 y$ works. I presume you want $S$ an open set or something. – Artimis Fowl Sep 06 '18 at 06:18
  • @ArtimisFowl $x \circ y = x^2 y$ is not associative: $(x\circ y)\circ z = x^4 y^2 z$, while $x\circ(y\circ z)=x^2 y^2 z$. – Danijel Sep 06 '18 at 11:36
  • it is on the given domain. $x^4 = x^2$ when $x$ is $\pm 1.$ there are only 4 inputs, try them. – Artimis Fowl Sep 06 '18 at 13:39
  • @ArtimisFowl You are right, I thought you wrote $[-1,1]^2$, my mistake. However, the output of this binary operation depends only on one argument because $x^2 = |x| = 1$ when restricted to ${-1, 1}$ and therefore $x\circ y = y$. – Danijel Sep 06 '18 at 13:50
  • You are right, so pick $S ={-1,0,1}.$ now if $x$ or $y$ is 0 matters, so it is not trivial. It's not commutative, analytic, and associative by previous work. But still unsatisfying I think? – Artimis Fowl Sep 06 '18 at 13:54
  • Technically, this answers the question the way it is posed. I should have added that $S$ shouldn't be "too small" subset of $\mathbb R \times \mathbb R$, perhaps that it shouldn't be a discrete set or that it should contain an open disk or that it should be connected. – Danijel Sep 06 '18 at 14:00

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