Ordering on Closure Operators iff Images are Including

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.

Let $f, g:S \to S$ be closure operators on $L$.

Then $f \preceq g$ $f\left[{S}\right] \subseteq g\left[{S}\right]$

where
 * $\preceq$ denotes the ordering on mappings,
 * $f\left[{S}\right]$ denotes the image of $f$.