Shelf

Remarks

Equivalently, for every $x \in X$ the left translation $y \mapsto x \triangleright y$ is an endomorphism of $(X, \triangleright)$. A shelf is a rack when every left translation is a bijection.

Example: let $X = \set{1, 2}$ with $1 \triangleright y = 2$ and $2 \triangleright y = y$ for all $y$; this is the Laver table $A_1$. If $x = 2$, both sides equal $y \triangleright z$. If $x = 1$, the left side is $2$, and the right side is $2 \triangleright 2 = 2$. So $X$ is a shelf; it is not a rack, since $y \mapsto 1 \triangleright y$ is not injective.

Non-example: $(\Z, +)$, since $x + (y + z) \ne (x + y) + (x + z)$ for $x \ne 0$.

Some authors use right self-distributivity instead, with $(z \triangleleft y) \triangleleft x = (z \triangleleft x) \triangleleft (y \triangleleft x)$; the two conventions are related by $x \triangleright y = y \triangleleft x$. In Mathlib this structure is the class Shelf.