TITLE: A binary operation, closed over the reals, that is associative, but not commutative
QUESTION [11 upvotes]: 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?

REPLY [2 votes]: Batominovski has another answer in the comments to this question. I will type up the checking:
Our candidate is $x\circ y=|x|y$. Then:

Associative? We have $(x\circ y)\circ z=(|x|y)\circ z=||x|y|z=|xy|z.$ On the other hand, $x\circ(y\circ z)=x\circ(|y|z)=|x||y|z=|xy|z.$
Non-commutative? $x\circ y=|x|y\not=|y|x=y\circ x$. 

So this solution of Batominovski's fits the bill.