Transitive Closure is Closure Operator
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Definition
Let $S$ be a set.
Let $\TT$ be the set of all endorelations on $S$.
Then the transitive closure operator is a closure operator on $\TT$.
Proof
Let $\QQ$ be the set of transitive relations on $S$.
By Intersection of Transitive Relations is Transitive, the intersection of any subset of $\QQ$ is in $\QQ$.
Recall the definition of transitive closure as the intersection of transitive supersets:
- The transitive closure of a relation $\RR$ on $S$ is the intersection of elements of $\QQ$ that contain $S$.
From Closure Operator from Closed Sets we conclude that transitive closure is a closure operator.
$\blacksquare$