Set System Closed under Intersection is Commutative Semigroup

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

Let $\mathcal S$ be such that:


 * $\forall A, B \in \mathcal S: A \cap B \in \mathcal S$

Then $\left({\mathcal S, \cap}\right)$ is a commutative semigroup.

Closure
By definition (above), $\left({\mathcal S, \cap}\right)$ is closed.

Associativity
The operation $\cap$ is associative from Intersection is Associative.

Commutativity
The operation $\cap$ is commutative from Intersection is Commutative.

Hence, by definition, the result.

Also see

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