Definition:Closure Operator/Ordering/Definition 2

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 $x, y \in S$:
 * $x \preceq \operatorname{cl} \left({y}\right) \iff \operatorname{cl} \left({x}\right) \preceq \operatorname{cl} \left({y}\right)$