# Power Set with Intersection is Commutative Monoid

## Theorem

Let $S$ be a set and let $\powerset S$ be its power set.

Then $\struct {\powerset S, \cap}$ is a commutative monoid whose identity is $S$.

The only invertible element of this structure is $S$.

Thus (except in the degenerate case $S = \O$) $\struct {\powerset S, \cap}$ cannot be a group.

## Proof

$\forall A, B \in \powerset S: A \cap B \in \powerset S$

From Set System Closed under Intersection is Commutative Semigroup, $\struct {\powerset S, \cap}$ is a commutative semigroup.

From Identity of Power Set with Intersection, we have that $S$ acts as the identity.

It remains to be shown that only $S$ has an inverse:

For $T \subseteq S$ to have an inverse under $\cap$, we require $T^{-1} \cap T = S$.

From this it follows that $T = S = T^{-1}$.

The result follows by definition of commutative monoid.

$\blacksquare$