Definition:Closure Operator/Ordering/Definition 1

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Let $\struct {S, \preceq}$ be an ordered set.

A closure operator on $S$ is a mapping:

$\cl: S \to S$

which satisfies the closure axioms as follows for all elements $x, y \in S$:

\((\text {cl} 1)\)   $:$   $\cl$ is inflationary:       \(\ds x \)   \(\ds \preceq \)   \(\ds \map \cl x \)      
\((\text {cl} 2)\)   $:$   $\cl$ is increasing:       \(\ds x \preceq y \)   \(\ds \implies \)   \(\ds \map \cl x \preceq \map \cl y \)      
\((\text {cl} 3)\)   $:$   $\cl$ is idempotent:       \(\ds \map \cl {\map \cl x} \)   \(\ds = \)   \(\ds \map \cl x \)      

Also see