Example of idempotent left quasigroups which are right-distributive but not left-distributive I am looking for examples of the following algebraic structure: a set (X,.)  which satisfy the axioms
(idempotent)  x.x = x
(left quasigroup)  the equation a.x = b has a unique solution denoted by x = a*b
(right distributive) (x ? y) ! z = (x ! z) ? (y ! z) , where  ? and ! are any of the operations . or *
but not the axiom
(left distributive) x ! (y ? z) = (x ! y) ? (x ! z)  for any choice of the operations ? and ! among . and *
Remark that an idempotent left quasigroup which is left distributive is called a quandle. The question can be rephrased as: give examples of idempotent left quasigroups which are right distributive but not quandles.
With the help of the (idempotent) axiom, the (right distributive) and (left distributive) may be rewritten as (right medial) and (left medial) axioms, so the question may be rephrased as: give (as many as possible) examples of idempotent left quasigroups which are right medial but not left medial.
The motivation for the question is described in the notes A problem concerning emergent algebras.
 A: We shall call an algebraic structure $(X,*_{0},*_{1})$ bi-right distributive if
$(x*_{i}y)*_{j}z=(x*_{j}z)*_{i}(y*_{j}z)$ whenever $x,y,z\in X,i,j\in\{0,1\}$, and we call
$(X,*_{0},*_{1})$ bi-left distributive if $x*_{i}(y*_{j}z)=(x*_{i}y)*_{j}(x*_{i}z)$ whenever $x,y,z\in X,i,j\in\{0,1\}$.
Suppose that $R$ is a commutative ring with unity and $M$ is a left $R$-module. Then for each $\lambda\in R$, define an operation $*_{\lambda}$ on $M$ by letting
$x*_{\lambda}y=(1-\lambda)x+\lambda y$. Then we satisfy the following identities:

*

*$x*_{\lambda}x=x$.


*$(x*_{\lambda}y)*_{\mu}z=(x*_{\mu}z)*_{\lambda}(y*_{\mu}z)$.


*$x*_{\lambda}(y*_{\mu}z)=(x*_{\lambda}y)*_{\mu}(x*_{\lambda}z)$.


*$x*_{\lambda}y=y*_{1-\lambda}x$.


*$x*_{\lambda\mu}y=x*_{\lambda}(x*_{\mu}y)$.


*$x*_{1}y=y$
Therefore, if $\lambda\in R$ and $\lambda$ is invertible, then $(M,*_{\lambda},*_{\lambda^{-1}})$ is a bi-left distributive, bi-right distributive idempotent left-quasigroup (and also a quandle). If $2\lambda=1$, then $*_{\lambda}$ is commutative, and if $\lambda^{2}=\lambda$, then $*_{\lambda}$ is associative. Therefore, if $(X,*,+)$ is a bi-right distributive idempotent left-quasigroup that is not left-distributive, then the direct product $(X,*,+)\times (M,*_{\lambda},*_{\lambda^{-1}})$ is also a bi-right distributive idempotent left-quasigroup that is not left-distributive. Therefore, $(M,*_{\lambda},*_{\lambda^{-1}})$ is a tool to produce bi-right distributive idempotent left-quasigroups that are not left-distributive.
