Power Set with Union is Commutative Monoid

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

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

The only invertible element of this structure is $\O$.

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

Proof
From Power Set is Closed under Union:
 * $\forall A, B \in \powerset S: A \cup B \in \powerset S$

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

From Identity of Power Set with Union, $\O$ acts as the identity of $\struct {\powerset S, \cup}$.

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

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

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

The result follows by definition of commutative monoid.

Also see

 * Power Set with Intersection is Commutative Monoid