Definition:Self-Distributive Operation

Definition

Let $\circ$ be a binary operation on the set $S$.

$\circ$ is self-distributive if and only if:

$(1): \quad \circ$ is left self-distributive

and:

$(2): \quad \circ$ is right self-distributive.

Left Self-Distributive

$\circ$ is left self-distributive if and only if:

$\forall a, b, c \in S: a \circ \paren {b \circ c} = \paren {a \circ b} \circ \paren {a \circ c}$

Right Self-Distributive

$\circ$ is right self-distributive if and only if:

$\forall a, b, c \in S: \paren {a \circ b} \circ c = \paren {a \circ c} \circ \paren {b \circ c}$

Also defined as

The term self-distributive operation is sometimes seen to be used for operations for which only one of the above holds.

Also see

• Self-Distributive Law for Conditional

• Results about self-distributive operations can be found here.