Very easy solution of my old conjecture

Like a complete idiot, this took me a few years to disprove my conjecture, despite the proof is quite trivial.

Here is the complete solution:

Example [S]\ne\{\bigsqcup^{\mathfrak{A}}X \mid X\in\mathscr{P} S\}, where [S] is the complete lattice generated by a strong partition S of filter on a set.

Proof Consider any infinite set U and its strong partition \{\uparrow^U\{x\} \mid x\in U\}.

\{\bigsqcup^{\mathfrak{A}}X \mid X\in\mathscr{P} S\} consists only of principal filters.

But [S] obviously contains some nonprincipal filters.

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