Structure Induced by Permutation on Semigroup is not necessarily Semigroup

Theorem
Let $\struct {S, \circ}$ be a semigroup.

Let $\sigma: S \to S$ be a permutation on $S$.

Let $\struct {S, \circ_\sigma}$ be the structure induced by $\sigma$ on $\circ$:
 * $\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$

Then $\struct {S, \circ_\sigma}$ is not necessarily itself a semigroup.

Proof
From Operation Induced by Permutation on Magma is Closed we have that $\struct {S, \circ_\sigma}$ is a closed structure.

Hence holds.

However, we have that Operation Induced by Permutation on Semigroup is not necessarily Associative.

Hence does not necessarily hold for $\struct {S, \circ_\sigma}$

Hence $\struct {S, \circ_\sigma}$ is not necessarily a semigroup.