Definition:Closure Operator/Ordering

Definition

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) $

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 $x, y \in S$:

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

$\operatorname{cl}$ 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	\(\displaystyle \operatorname{cl} \left({\operatorname{cl} \left({x}\right)}\right) \)	\(\displaystyle = \)	\(\displaystyle \operatorname{cl} \left({x}\right) \)