What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

Let $\mathfrak{A}$ be an indexed family of sets.

\emph{Products} are $\prod A$ for $A \in \prod \mathfrak{A}$.

\emph{Hyperfuncoids} are filters $\mathfrak{F} \Gamma$ on the lattice $\Gamma$ of all finite unions of products.

\begin{problem} Is $\bigcap^{\mathsf{\tmop{FCD}}}$ a bijection from hyperfuncoids $\mathfrak{F} \Gamma$ to: \begin{enumerate} \item prestaroids on $\mathfrak{A}$; \item staroids on $\mathfrak{A}$; \item completary staroids on $\mathfrak{A}$? \end{enumerate} If yes, is $\operatorname{up}^{\Gamma}$ defining the inverse bijection? If not, characterize the image of the function $\bigcap^{\mathsf{\tmop{FCD}}}$ defined on $\mathfrak{F} \Gamma$.

Consider also the variant of this problem with the set $\Gamma$ replaced with the set $\Gamma^{\ast}$ of complements of elements of the set $\Gamma$. \end{problem}

Keywords: hyperfuncoids; multidimensional

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