Set System Closed under Intersection is Commutative Semigroup

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.

$\blacksquare$

Also see

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