Definition:Closure Operator/Ordering

Definition

Definition 1

Let $\left({S, \preceq}\right)$ be an ordered set.

A closure operator on $S$ is a mapping:

$\operatorname{cl}: S \to S$

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

		$\operatorname{cl}$ is inflationary			\(\displaystyle x \)	\(\displaystyle \preceq \)	\(\displaystyle \operatorname{cl} \left({x}\right) \)
		$\operatorname{cl}$ is increasing			\(\displaystyle x \preceq y \)	\(\displaystyle \implies \)	\(\displaystyle \operatorname{cl} \left({x}\right) \preceq \operatorname{cl} \left({y}\right) \)
		$\operatorname{cl}$ is idempotent			\(\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({x}\right)}\right) \)	\(\displaystyle = \)	\(\displaystyle \operatorname{cl} \left({x}\right) \)

Definition 2

Let $\left({S, \preceq}\right)$ be an ordered set.

A closure operator on $S$ is a mapping:

$\operatorname{cl}: S \to S$

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

$x \preceq \operatorname{cl} \left({y}\right) \iff \operatorname{cl} \left({x}\right) \preceq \operatorname{cl} \left({y}\right)$

Also see

• Equivalence of Definitions of Closure Operator
• Results about closure operators can be found here.