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What are the binary functions $F$ of the real numbers, possibly taking an open subset or including infinity, that have an identity element and a zero element and are associative? I know that $F:[-\infty,\infty)^2\to\mathbb R, F(x,y)=x+y+m$ works with identity $-m$ and idempotent $-\infty$, and similarly $F:{\mathbb R}^2\to\mathbb R , F(x,y)=mxy$, but are there any others?

1Rock
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    Take any bijection $\phi:\mathbb R\to\mathbb R$ and define $F(x,y)=\phi^{-1}(\phi(x)+\phi(y))$. (Also, instead of adding you can use multiplication.) Yours are two special cases $\phi(x)=x+m$ and $\phi(x)=mx$ (for $m\ne 0$). –  Mar 14 '23 at 22:05
  • Thanks! Do you know if there are any others, or if those cases are the only nontrivial ones? – 1Rock Mar 15 '23 at 12:01
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    Yes. The most general you can go is this: (a) Take a set $S$ of cardinality $|\mathbb R|=\mathfrak c=2^{\aleph_0}$. (b) Equip it with a structure of a monoid, to become $(S,\circ)$ (c) Take any bijection $\phi:\mathbb R\to S$. (d) Define $x\circ y:=\phi^{-1}(\phi(x)\circ\phi(y))$. Non-isomorphic monoids of size $\mathfrak c$ would give you substantially different structures on $\mathbb R$. For example, carry $\oplus$ ("xor" operation) on $2={0,1}$ onto $2^{\aleph_0}$ component-wise and use any bijection of $\mathbb R$ to $2^{\aleph_0}$. –  Mar 15 '23 at 17:08
  • (The above monoid is substantially different from both examples you are mentioning because (for its zero element $e$) it satisfies $x\oplus x=e$ for every element $x\in\mathbb R$ i.e. it is a torsion monoid.) Another (somewhat degenerate) case is to choose any $e\in\mathbb R$ and define $x\circ y:=e$. –  Mar 15 '23 at 17:12
  • That's great, thanks! How do you know that's the most general you can go? – 1Rock Mar 16 '23 at 00:00
  • Because for any such monoid $(\mathbb R,\circ)$ it is of that form for $S=\mathbb R, \phi=Id_{\mathbb R}$ –  Mar 16 '23 at 07:23

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