# Power Structure of Semigroup is Semigroup

## Theorem

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

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.

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

## Proof

From Power Structure of Magma 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.

$\blacksquare$