Structure Induced by Semigroup Operation is Semigroup

Theorem
Let $\struct {T, \odot}$ be a semigroup whose identity is $e$.

Let $S$ be a set.

Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\circ$.

Then $\struct {T^S, \oplus}$ is a semigroup.

Proof
Taking the semigroup axioms in turn:

As $\struct {T, \circ}$ is a semigroup, it is closed by.

From Closure of Pointwise Operation on Algebraic Structure it follows that $\struct {T^S, \oplus}$ is likewise closed.

As $\struct {T, \circ}$ is a semigroup, $\circ$ is associative.

So from Structure Induced by Associative Operation is Associative, $\struct {T^S, \oplus}$ is also associative.

All the semigroup axioms are thus seen to be fulfilled, and so $\struct {T^S, \oplus}$ is a semigroup.