Definition:Closure Operator/Power Set

Definition

Let $S$ be a set.

Let $\powerset S$ denote the power set of $S$.

A closure operator on $S$ is a mapping:

$\cl: \powerset S \to \powerset S$

which satisfies the closure axioms as follows for all sets $X, Y \subseteq S$:

\((\text {cl} 1)\)	$:$		$\cl$ is inflationary:	\(\ds \forall X \subseteq S:\)		\(\ds X \)	\(\ds \subseteq \)	\(\ds \map \cl X \)
\((\text {cl} 2)\)	$:$		$\cl$ is increasing:	\(\ds \forall X, Y \subseteq S:\)		\(\ds X \subseteq Y \)	\(\ds \implies \)	\(\ds \map \cl X \subseteq \map \cl Y \)
\((\text {cl} 3)\)	$:$		$\cl$ is idempotent:	\(\ds \forall X \subseteq S:\)		\(\ds \map \cl {\map \cl X} \)	\(\ds = \)	\(\ds \map \cl X \)

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.25$

although at this point he does not name this operator, just describes it