Definition:Closure Operator/Ordering/Definition 2

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