Left Operation is Distributive over Idempotent Operation

Theorem
Let $\left({S, \circ, \leftarrow}\right)$ be an algebraic structure where $\leftarrow$ is the left operation and $\circ$ is any arbitrary binary operation.

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

Proof
From Left Operation is Right Distributive over All Operations we have that:
 * $\forall a, b, c \in S: \left({a \circ c}\right) \leftarrow c = \left({a \leftarrow c}\right) \circ \left({b \leftarrow c}\right)$

for all binary operations $\circ$.

It remains to show that $\leftarrow$ is left distributive over $\circ$ iff $\circ$ is idempotent.

Necessary Condition
Suppose $\circ$ is idempotent.

Then by definition of the left operation:

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

Sufficient Condition
Suppose $\leftarrow$ is left distributive over $\circ$.

Let $a \in S$ be arbitrary.

Then:

Hence $\circ$ is idempotent.