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.
$\blacksquare$
