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 $$*$$.