Operation over which Every Commutative Associative Operation is Distributive is either Left or Right Operation

Theorem
Let $\struct {S, \circ}$ be an algebraic structure.

Let $\circ$ be such that every operation on $S$ which is both commutative and associative is distributive over $\circ$.

Then $\circ$ is either the left operation $\gets$ or the right operation $\to$.

Proof
Recall from:
 * Every Operation is Distributive over Right Operation

and:
 * Every Operation is Distributive over Left Operation

that if $\circ$ is either $\gets$ or $\to$, then every operation is distributive over it, whether commutative or associative.

Lemma
Hence $\odot$ must be distributive over $\circ$.

First note that we have:

But we have that $\odot$ is distributive over $\circ$.

That is:
 * $\forall x, y, z \in S: x \odot \paren {y \circ z} = \paren {x \odot z} \circ \paren {y \odot z}$

and so as $a$ and $b$ are arbitrary:
 * $\forall a, b \in S: \set {a \circ a, a \circ b, b \circ a, b \circ b} = \set {a, b}$

This of course also applies to:
 * $y \circ z \in \set {y, z}$

and:
 * $x \circ y \in \set {x, y}$

As $a$ and $b$ are arbitrary, the result applies for all $a, b \in S$.