Definition:Closure Operator/Ordering/Definition 2

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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 condition for all elements $x, y \in S$:

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