\begin{definition}[Definition:Closure Operator/Ordering/Definition 2]
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 $x, y \in S$:
:$x \preceq \map \cl y \iff \map \cl x \preceq \map \cl y$
\end{definition}