# Right Operation is Distributive over Idempotent Operation

## Theorem

Let $\struct {S, \circ, \rightarrow}$ be an algebraic structure where:

$\rightarrow$ is the right operation
$\circ$ is any arbitrary binary operation.

Then:

$\rightarrow$ is distributive over $\circ$
$\circ$ is idempotent.

## Proof

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

for all binary operations $\circ$.

It remains to show that $\rightarrow$ is right distributive over $\circ$ if and only if $\circ$ is idempotent.

### Necessary Condition

Let $\circ$ be idempotent.

Then:

 $\displaystyle \paren {a \circ b} \rightarrow c$ $=$ $\displaystyle c$ Definition of Right Operation $\displaystyle$ $=$ $\displaystyle c \circ c$ Definition of Idempotent Operation $\displaystyle$ $=$ $\displaystyle \paren {a \rightarrow c} \circ \paren {b \rightarrow c}$ Definition of Right Operation

Thus $\rightarrow$ is right distributive over $\circ$.

$\Box$

### Sufficient Condition

Let $\rightarrow$ be right distributive over $\circ$.

Let $c \in S$ be arbitrary.

Then:

 $\displaystyle c$ $=$ $\displaystyle \paren {a \circ b} \rightarrow c$ Definition of Right Operation $\displaystyle$ $=$ $\displaystyle \paren {a \rightarrow c} \circ \paren {b \rightarrow c}$ Definition of Right Distributive Operation $\displaystyle$ $=$ $\displaystyle c \circ c$ Definition of Right Operation

Hence $\circ$ is idempotent.

$\blacksquare$