Axiom:Closure Axioms

Definition

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

A closure operator on $S$ is a mapping $\cl: S \to S$ satisfying the following 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:			\(\ds \map \cl {\map \cl x} \)	\(\ds = \)	\(\ds \map \cl x \)