A conjecture about direct product of funcoids

Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Topology
Recomm. for undergrads: no
Posted by: porton
on: July 26th, 2012

\begin{conjecture} Let $f_1$ and $f_2$ are monovalued, entirely defined funcoids with $\operatorname{Src}f_1=\operatorname{Src}f_2=A$. Then there exists a pointfree funcoid $f_1 \times^{\left( D \right)} f_2$ such that (for every filter $x$ on $A$) $$\left\langle f_1 \times^{\left( D \right)} f_2 \right\rangle x = \bigcup \left\{ \langle f_1\rangle X \times^{\mathsf{FCD}} \langle f_2\rangle X \hspace{1em} | \hspace{1em} X \in \mathrm{atoms}^{\mathfrak{A}} x \right\}.$$ (The join operation is taken on the lattice of filters with reversed order.) \end{conjecture}

A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.

See \href [Algebraic General Topology]{http://www.mathematics21.org/algebraic-general-topology.html} for definitions of used concepts. % 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/}


*Victor Porton. \href [a blog post]{http://portonmath.wordpress.com/2012/07/26/conjecture-direct-product-funcoids/} % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.