A binary operation, closed over the reals, that is associative, but not commutative

Question

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that holds this property? And can you give a specific example?

Answer 1

We can define $x \oplus y=y$. Then $(x \oplus y) \oplus z =z= x \oplus (y \oplus z)$ but $y=x \oplus y \neq y \oplus x=x$

Answer 2

Yes, there are many other bilinear products on a given vector space which are associative but not commutative. For example, real associative algebras (not necessarily commutative), see also this MO-question.

Furthermore the quaternion algebra is a real division algebra which is associative but not commutative.

Edit: You added that the algebra must be part of the real numbers. Then also your example with the matrix algebra does no longer work.

Answer 3

If you already know that matrix multiplication is associative, but not commutative, then you can just choose your favorite bijection $f:\mathbb R\rightarrow M_2(\mathbb R)$, since the two sets are equinumerous. Then, define $a\oplus b = f^{-1}(f(a)f(b))$ to get an operation on $\mathbb R$ which is associative, but not commutative. If you want to have inverses as well, then you can replace $f$ with your favorite map from $\mathbb R$ to the invertible $2\times 2$ matrices.

In general, without more structure, you are equivalently asking, "Is there any associative, but not commutative operation defined on a domain of cardinality $|\mathbb R|$?" since the role of $\mathbb R$ in the question is nothing more than a set.