What are some examples of infinite strict quasigroups? 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:


*

*Infinite strict quasigroups that are idempotent

*Infinite strict quasigroups with neither a left nor a right identity

*Infinite strict quasigroups that are idempotent and have neither a left nor a right identity


But other examples "interesting" infinite strict quasigroups are welcome!
 A: 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.


*

*In fact, this has a strong "nonidentity" property: if $x\not=y$ then $x*y\not\in\{x,y\}$. This is the strongest failure of the existence of an identity element we can possibly have in an idempotent quasigroup.



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 $\overline{xz}$.
A: 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:


*

*Etherington, I. (1965). Quasigroups and cubic curves. Proceedings of the Edinburgh Mathematical Society, 14(4), 273-291. doi:10.1017/S001309150000897X

*Yu. I. Manin, Cubic Forms, Algebra, Geometry, Arithmetic, Second Edition, North-Holland, 1986

A: The midpoint (or mean) between two points in $\mathbb{R}^n$.
$(a+b)/2=b$ iff $a=b$.
