# 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 $*$.

The operation $\circ$ is distributive over $*$, or distributes over $*$, if and only if:

$\circ$ is right distributive over $*$

and:

$\circ$ is left distributive over $*$.

### Left Distributive

The operation $\circ$ is left distributive over the operation $*$ if and only if:

$\forall a, b, c \in S: a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}$

### Right Distributive

The operation $\circ$ is right distributive over the operation $*$ if and only if:

$\forall a, b, c \in S: \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}$

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

### Distributand

Let $\circ$ be distributive over $*$.

Then $*$ is a distributand of $\circ$.

### Distributor

Let $\circ$ be distributive over $*$.

Then $\circ$ is a distributor of $*$.

## Also see

• Results about distributive operations can be found here.