# 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.