Compositions of Closure Operators are both Closure Operators iff Operators Commute

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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 if and only if it is idempotent.

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

$\blacksquare$