Operation Induced by Permutation on Semigroup is not necessarily Associative

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

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

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

Then $\circ_\sigma$ is not necessarily associative on $S$.

Proof

 * Proof by Counterexample

Let $S = \set {a, b, c}$.

Let $\circ$ denote the right operation on $S$:
 * $\forall x, y \in S: x \to y = y$

By Right Operation is Associative we have that $\circ$ is an associative operation, and is trivially closed on $S$.

Hence $\struct {S, \circ}$ is a semigroup, and we have:


 * $a \circ \paren {b \circ c} = c = \paren {a \circ b} \circ c$

Let $\sigma$ denote the permutation on $S$ defined as:

We have:

Then:

So $a \circ_\sigma \paren {b \circ_\sigma c} \ne \paren {a \circ_\sigma b} \circ_\sigma c$ and the result follows.