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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.9 \ \text {(a)}$