Definition:Self-Distributive Operation

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

Then $\circ$ is self-distributive iff:


 * $\forall a, b, c \in S: \left({a \circ b}\right) \circ c = \left ({a \circ c}\right) \circ \left({b \circ c}\right)$
 * $\forall a, b, c \in S: a \circ \left({b \circ c}\right) = \left({a \circ b}\right) \circ \left({a \circ c}\right)$

The term is sometimes used for operations for which only one of the above holds.

The logical connective "$\implies$" (conditional) is one of those - see Self-Distributive Law for Conditional.