Definition:Closure Operator/Ordering

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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 $x, y \in S$:

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


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