![](/files/happy5.png)
A conjecture about direct product of funcoids
Conjecture Let
and
are monovalued, entirely defined funcoids with
. Then there exists a pointfree funcoid
such that (for every filter
on
)
(The join operation is taken on the lattice of filters with reversed order.)
![$ f_1 $](/files/tex/472cbb02b9334d02f91ee6d85190987d38d55395.png)
![$ f_2 $](/files/tex/0470e2ed22fecd48d8ad613016446ca8d5540085.png)
![$ \operatorname{Src}f_1=\operatorname{Src}f_2=A $](/files/tex/74804e1f464f0ed9acb65fbbe676a40dc27a919b.png)
![$ f_1 \times^{\left( D \right)} f_2 $](/files/tex/0a40b9e9b410bc22dbd5b368fbadae778feb7e62.png)
![$ x $](/files/tex/e7ba5befcaa0d78e43b5176d70ce67425fd0fcdc.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$$\left\langle f_1 \times^{\left( D \right)} f_2 \right\rangle x = \bigcup \left\{ \langle f_1\rangle X \times^{\mathsf{FCD}} \langle f_2\rangle X \hspace{1em} | \hspace{1em} X \in \mathrm{atoms}^{\mathfrak{A}} x \right\}.$$](/files/tex/2326592ad6b621142c337b6acc2a4b724ca723f4.png)
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.
Bibliography
*Victor Porton. a blog post
* indicates original appearance(s) of problem.