Self-distributive operation

Remarks

Left self-distributivity says that $*$ has the distributive property over itself from the left. Equivalently, each left translation $L_x \colon y \mapsto x * y$ is an endomorphism of $(X, *)$.

Example: on $\Z$ let $x * y = 2x - y$. Then $x * (y * z) = 2x - (2y - z) = 2x - 2y + z$ and $(x * y) * (x * z) = 2(2x - y) - (2x - z) = 2x - 2y + z$, so $*$ is left self-distributive.

Example: in a group, $x * y = x y x^{-1}$ is left self-distributive, since both sides equal $x y z y^{-1} x^{-1}$.

Non-example: addition on $\Z$. Here $x + (y + z) = x + y + z$ while $(x + y) + (x + z) = 2x + y + z$, which differ when $x \ne 0$.