# Power Structure of Monoid is Monoid

## Theorem

Let $\struct {G, \circ}$ be a monoid with identity $e$.

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

Then $\struct {\powerset G, \circ_\PP}$ is a monoid with identity $\set e$.

## Proof

By definition of a monoid, $\struct {G, \circ}$ is a semigroup.

By Power Structure of Semigroup is Semigroup, $\struct {\powerset G, \circ_\PP}$ is a semigroup.

By Subset Product by Identity Singleton, $\set e$ is an identity for $\struct {\powerset G, \circ_\PP}$.

Thus $\struct {\powerset G, \circ_\PP}$ is a monoid with identity $\set e$.

$\blacksquare$