Definition:Distributive Operation

Definition
Let $S$ be a set on which is defined two binary operations, defined on all the elements of $S \times S$, which we will denote as $\circ$ and $*$.

Left Distributive
The operation $\circ$ is left distributive over the operation $*$ iff:


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

Right Distributive
The operation $\circ$ is right distributive over the operation $*$ iff:


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

Distributive
If $\circ$ is both right and left distributive over $*$, then $\circ$ is distributive over $*$, or $\circ$ distributes over $*$.

Distributand and Distributor
So as to streamline what may turn into cumbersome language, some further definitions:

If $\circ$ is distributive over $*$, then $*$ is a distributand of $\circ$, and $\circ$ is a distributor of $*$.