Groupoid with division I'm searching an example of a grupoid with division which is not a quasigroup. A grupoid $(G, \cdot)$ is with division if $a\cdot G=G\cdot a=G$. I was thinking to try $(\mathbb{Q},\cdot)$, where $x\cdot y=|x-y|$. It is easy to see that it isn't a quasigroup and $a \cdot \mathbb{Q} \subset \mathbb{Q}$. I tried the other inclusion and didn't get the result. I'm sure I am missing something.

In case it may still be useful, I advise you to have a look at this paper, in particular example 2 at page 44. Here several examples are shown, over finite and infinite sets.
Notice that left and right division, together with left or right cancellation only (or viceversa), make the infinite cardinality matter.
