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Let $\mathfrak{F}(S)$ denotes the set of filters (including the improper filter) on a poset $S$, ordered reversely to set theoretic inclusion of filters. Let $Da$ for a lattice element $a$ denote its sublattice $\{ x \mid x \leq a \}$. Let $Z(X)$ denotes the set of complemented elements of the lattice $X$.

Conjecture $\mathfrak{F}(Z(D\mathcal{A}))$ is order-isomorphic to $D\mathcal{A}$ for every filter $\mathcal{A}$ on a set $U$ (and the lattice $\mathfrak{F}(\mathscr{P}U)$). If they are isomorphic, find an isomorphism.

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I have proved this conjecture.

See my book. Currently it is theorem number 598.

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