Set System Closed under Union is Commutative Semigroup

Theorem
Let $\mathcal S$ be a system of sets.

Let $\mathcal S$ be such that:
 * $\forall A, B \in \mathcal S: A \cup B \in \mathcal S$

Then $\struct {\mathcal S, \cup}$ is a commutative semigroup.

Closure
By definition (above), $\struct {\mathcal S, \cup}$ is closed.

Associativity
The operation $\cup$ is associative from Union is Associative.

Commutativity
The operation $\cup$ is commutative from Union is Commutative.

Hence, by definition, the result.

Also see

 * Set System Closed under Intersection is Commutative Semigroup
 * Set System Closed under Symmetric Difference is Abelian Group