A new funcoid discovered

It is easy to prove that the equation \langle \mathscr{A} \rangle X = \mathrm{atoms}^{\mathfrak{A}}\, X (for principal filters X) defines a (unique) funcoid \mathscr{A} which I call quasi-atoms funcoid.

Note that as it is easy to prove \langle \mathscr{A}^{-1} \rangle Y = \bigsqcup Y for every set Y of ultrafilters.

Does this funcoid posses interesting properties? Can it be used to prove any open problem?

What is its behavior on non-principal filters?

I started researching properties of this weird funcoid in this PDF file.

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