Definition:Distributive Operation

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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 $*$.


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


Left Distributive

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

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


Right Distributive

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

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


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.


Sources