Power Structure of Semigroup is Semigroup

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

Let $\struct {\powerset S, \circ_\PP}$ be the algebraic structure consisting of the power set of $S$ and the operation induced on $\powerset S$ by $\circ$.

Then $\struct {\powerset S, \circ_\PP}$ is a semigroup.

Proof
From Power Set of Magma under Induced Operation is Magma we conclude that $\struct {\powerset S, \circ_\PP}$ is a magma.

It follows from Subset Product within Semigroup is Associative that $\circ_\PP$ is associative in $\struct {\powerset S, \circ_\PP}$.

Thus $\struct {\powerset S, \circ_\PP}$ is a semigroup.