Groupoid with division

Question

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.

Answer 1

Define the following operation $*$ on ${\bf N}$ for each $n,m$, if $n\leq m$, define $n*m=m*n=\lvert m-n\rvert$. Then clearly $n*{\bf N}={\bf N}*n={\bf N}$, but $0*1=2*1=1$.

Doing it with ${\bf Q}$ would not work because it contains negative elements. But if you restricted to nonnegative rationals (or, in fact, the nonnegative elements in any ordered abelian group), the same example would work (although I think it would be less clear than with the natural numbers).

Answer 2

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.