Set System Closed under Union is Commutative Semigroup

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Theorem

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

Let $\mathcal S$ be such that:

$\forall A, B \in \mathcal S: A \cup B \in \mathcal S$


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


Proof

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.

$\blacksquare$


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