Set System Closed under Intersection is Commutative Semigroup

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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.


Proof

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.

$\blacksquare$


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