Definition:Closure Operator/Ordering/Definition 2

Definition
Let $\struct {S, \preceq}$ be an ordered set.

A closure operator on $S$ is a mapping:
 * $\cl: S \to S$

which satisfies the following condition for all elements $x, y \in S$:
 * $x \preceq \map \cl y \iff \map \cl x \preceq \map \cl y$

Also see

 * Equivalence of Definitions of Closure Operator