Definition:Closure Operator/Power Set

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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 following conditions for all sets $X, Y \subseteq S$:

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


A closure operator on a set $S$ in this sense is a closure operator on the power set of that set under the order-theoretic definition. In the unlikely case that these senses of "on" lead to an ambiguity, it should be resolved in the text.