Structure Induced by Permutation on Semigroup is not necessarily Semigroup

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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 Semigroup Axiom $\text S 0$: Closure holds.

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

Hence Semigroup Axiom $\text S 1$: Associativity does not necessarily hold for $\struct {S, \circ_\sigma}$

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

$\blacksquare$


Sources