# What are hyperfuncoids isomorphic to?

 Importance: Medium ✭✭
 Author(s): Porton, Victor
 Subject: Topology
 Keywords: hyperfuncoids multidimensional
 Posted by: porton on: December 9th, 2014

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}

It's used notation from \href[Algebraic General Topology draft book]{http://www.mathematics21.org/algebraic-general-topology.html}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

## Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.