# 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.