Compositions of Closure Operators are both Closure Operators iff Operators Commute

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $f$ and $g$ be closure operators on $S$.

Then the following are equivalent:


 * $(1): \quad f \circ g$ and $g \circ f$ are both closure operators.
 * $(2): \quad f$ and $g$ commute (that is, $f \circ g = g \circ f$).
 * $(3): \quad \operatorname{img}\left({f \circ g}\right) = \operatorname{img}\left({g \circ f}\right)$

where $\operatorname{img}$ represents the image of a mapping.

Proof
By Composition of Inflationary Mappings is Inflationary:


 * $f \circ g$ and $g \circ f$ are inflationary.

By Composition of Increasing Mappings is Increasing:


 * $f \circ g$ and $g \circ f$ are increasing.

Thus each of the two composite mappings will be a closure operator iff it is idempotent.

Therefore the equivalences follow from Composition of Inflationary and Idempotent Mappings.