By strict quasigroup I mean a quasigroup with no identity. I've come across one so far in the answer to this question, but I can't seem to find any others. I am particularly interested in finding example of:
But other examples "interesting" infinite strict quasigroups are welcome!
On
Another example of quasigroup with infinite underlying set and no identity element is given by the set of the points of a plane cubic curve, with the operation $$a\circ b=c$$ where $a$, $b$ and $c$ are the three intersections of the curve with a same straight line.
No identity: it does not exist any point $e$ suche that $e\circ a=a$ or either $a\circ e=a$ no matter what other point $a$ is.
But, this operation is totally symmetric: if $$a\circ b=c,$$ then $$\sigma(a)\circ\sigma(b)=\sigma(c)$$ where $\sigma$ is any permutation of $(a,b,c)$.
Two references to it:
At a glance, I think the operation $$x*y=2y-x$$ on $\mathbb{R}$ gives an infinite strict idempotent quasigroup with no left or right identity.
This can of course be generalized to $\mathbb{R}^n$ (or indeed a wide class of metric spaces): set $x*y$ to be the unique point $z$ such that $y$ is the midpoint of the line segment $\overline{xz}$.