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, $f \circ g = g \circ f$).

$(3): \quad \Img {f \circ g} = \Img {g \circ f}$

where $\Img \cdot$ denotes the image of a mapping.

Proof

By Composition of Inflationary Mappings is Inflationary:

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

By Composite 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.

$\blacksquare$