# Set System Closed under Intersection is Commutative Semigroup

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## Theorem

Let $\SS$ be a system of sets.

Let $\SS$ be such that:

- $\forall A, B \in \SS: A \cap B \in \SS$

Then $\struct {\SS, \cap}$ is a commutative semigroup.

## Proof

### Closure

We have by hypothesis that $\struct {\SS, \cap}$ 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.

$\blacksquare$