Left Operation is Distributive over Idempotent Operation

Theorem

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

$\leftarrow$ is the left operation
$\circ$ is any arbitrary binary operation.

Then:

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

Proof

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

for all binary operations $\circ$.

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

Necessary Condition

Let $\circ$ be idempotent.

Then:

 $\displaystyle a \leftarrow \paren {b \circ c}$ $=$ $\displaystyle a$ Definition of Left Operation $\displaystyle$ $=$ $\displaystyle a \circ a$ Definition of Idempotent Operation $\displaystyle$ $=$ $\displaystyle \paren {a \leftarrow b} \circ \paren {a \leftarrow c}$ Definition of Left Operation

Thus $\leftarrow$ is left distributive over $\circ$.

$\Box$

Sufficient Condition

Let $\leftarrow$ be left distributive over $\circ$.

Let $a \in S$ be arbitrary.

Then:

 $\displaystyle a$ $=$ $\displaystyle a \leftarrow \paren {b \circ c}$ Definition of Left Operation $\displaystyle$ $=$ $\displaystyle \paren {a \leftarrow b} \circ \paren {a \leftarrow c}$ Definition of Left Distributive Operation $\displaystyle$ $=$ $\displaystyle a \circ a$ Definition of Left Operation

Hence $\circ$ is idempotent.

$\blacksquare$