Definition:Closure Operator/Ordering

Definition

Definition 1

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:			\(\ds \map \cl {\map \cl x} \)	\(\ds = \)	\(\ds \map \cl x \)

Definition 2

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

A closure operator on $S$ is a mapping:

$\cl: S \to S$

which satisfies the following condition for all elements $x, y \in S$:

$x \preceq \map \cl y \iff \map \cl x \preceq \map \cl y$

Also see

• Equivalence of Definitions of Closure Operator

• Results about closure operators can be found here.