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 $*$.

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

Also see
So as to streamline what may turn into cumbersome language, some further definitions: