Transitive Closure is Closure Operator

Definition
Let $S$ be a set.

Let $\mathscr A$ be the set of all endorelations on $S$.

Then the transitive closure operator is a closure operator on $\mathscr A$.

Proof
Let $\mathcal Q$ be the set of transitive relations on $S$.

By Intersection of Transitive Relations is Transitive, the intersection of any subset of $\mathcal Q$ is in $\mathcal Q$.

Recall the definition of transitive closure as the intersection of transitive supersets:


 * The transitive closure of a relation $\mathcal R$ on $S$ is the intersection of elements of $\mathcal Q$ that contain $S$.

From Closure Operator from Closed Sets we conclude that transitive closure is a closure operator.