Structure Induced by Semigroup Operation is Semigroup

Theorem

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

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:

Semigroup Axiom $\text S 0$: Closure

As $\struct {T, \circ}$ is a semigroup, it is closed by Semigroup Axiom $\text S 0$: Closure.

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

$\Box$

Semigroup Axiom $\text S 1$: Associativity

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

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

$\Box$

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

$\blacksquare$