I have proved this recently formulated conjecture. See my book. Currently it is theorem number 598.

read moreLet $latex \mathfrak{F}(S)$ denotes the set of filters on a poset $latex S$, ordered reversely to set theoretic inclusion of filters. Let $latex Da$ for a lattice element $latex a$ denote its sublattice $latex \{ x \mid x \leq a \}$. Let…

read moreLike 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 $latex [S]\ne\{\bigsqcup^{\mathfrak{A}}X \mid X\in\mathscr{P} S\}$, where $latex [S]$ is the complete lattice generated by a strong partition $latex…

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