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
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.)

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

See Algebraic General Topology for definitions of used concepts.


*Victor Porton. a blog post

* indicates original appearance(s) of problem.