Definition:Self-Distributive Operation

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 "$$\Longrightarrow$$" (conditional) is one of those - see Implication Properties.