Operation on Structure of Cardinality 4+ where Every Permutation is Automorphism is Left or Right Operation

Theorem
Let $S$ be a set whose cardinality is at least $4$.

Let $\struct {S, \circ}$ be an algebraic structure on $S$ such that every permutation on $S$ is an automorphism on $\struct {S, \circ}$.

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

Proof
From Structure of Cardinality 3+ where Every Permutation is Automorphism is Idempotent, we have that $\circ$ is idempotent:
 * $\forall a \in S: a \circ a = a$

$\circ$ is such that:
 * $x \circ y = z$

for some distinct $x, y, z \in S$.

As $S$ has cardinality of at least $4$, there exists $w \in S$ such that $w \ne x$, $w \ne y$ and $w \ne z$.

Let $f$ be a permutation on $S$ such that:
 * $\map f x = x$
 * $\map f y = y$
 * $\map f z = w$

Then:

This contradicts our assertion that $w$ and $z$ are distinct.

Hence we have shown that:
 * $\forall x, y \in S: x \circ y = x \lor x \circ y = y$

$\circ$ is neither the left operation nor the right operation.

From the above, that means there exist $w, x, y, z \in S$ such that:

Let $f$ be a permutation on $S$ such that:

Then we have:

This contradicts our assertion that $w$ and $z$ are distinct.

Hence $\circ$ is either the left operation or the right operation.