nLab self-distributive operation

Definition

Let $\cdot\colon M \times M\to M$ be a binary operation, i.e. $(M,\cdot)$ is a magma. We say that the operation $\cdot$ is

• left self-distributive if for all $x,y,z\in M$, $x\cdot(y\cdot z) = (x\cdot y)\cdot (x\cdot z)$;
• right self-distributive if for all $x,y,z\in M$, $(y\cdot z)\cdot x = (y\cdot x)\cdot (z\cdot x)$.

See also shelf.

Examples

• The binary operation in any semilattice is self-distributive on both sides, following from associativity, commutativity, and idempotence.

• The operations in a rack (and hence also in a quandle) are self-distributive on the side on which they act. In particular, this includes the operation of conjugation in a group.

• A Laver table is the multiplication table of a self-distributive operation.