This MSE question asks about binary operations on the real numbers which are associative, but not commutative. Two answers are given:
- The operation $\circ$ defined by $x \circ y=x$.
- Letting $f:\mathbb R\to M_n(\mathbb R)$ be a bijection, then $x*y=f^{-1}(f(x)\cdot f(y))$, where $\cdot$ is matrix multiplication.
Operation 1 is good, but is what I would call a trivial binary operation as it only depends on one of its inputs. Operation 2 is far from satisfying since it does not respect the structure of the reals at all. So my question is,
Does there exist a binary operation $\star$ on the real numbers which is
- associative,
- non-commutative,
- nontrivial (operators of the form $x\circ y=f(x)$ or $x\circ y=g(y)$ are trivial), and
- continuous (with respect to the usual topologies on $\mathbb R^2$ and $\mathbb R$)?
I would also be satisfied with an operator where condition 4 was relaxed to
4'. continuous almost everywhere?
Best Answer
Ok, Batominovski's got an answer in the comments. I will type up the checking:
Our candidate is $x\circ y=|x|y$. Then:
So this solution of Batominovski's fits the bill.