Open Problem Garden - A conjecture about direct product of funcoids - Comments http://openproblemgarden.org/op/a_conjecture_about_direct_product_of_funcoids Comments for "A conjecture about direct product of funcoids" en A conjecture about direct product of funcoids http://openproblemgarden.org/op/a_conjecture_about_direct_product_of_funcoids <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/porton_victor">Porton</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/topology">Topology</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/472cbb02b9334d02f91ee6d85190987d38d55395.png" alt="$ f_1 $" /> and <img class="teximage" src="/files/tex/0470e2ed22fecd48d8ad613016446ca8d5540085.png" alt="$ f_2 $" /> are monovalued, entirely defined funcoids with <img class="teximage" src="/files/tex/74804e1f464f0ed9acb65fbbe676a40dc27a919b.png" alt="$ \operatorname{Src}f_1=\operatorname{Src}f_2=A $" />. Then there exists a pointfree funcoid <img class="teximage" src="/files/tex/0a40b9e9b410bc22dbd5b368fbadae778feb7e62.png" alt="$ f_1 \times^{\left( D \right)} f_2 $" /> such that (for every filter <img class="teximage" src="/files/tex/e7ba5befcaa0d78e43b5176d70ce67425fd0fcdc.png" alt="$ x $" /> on <img class="teximage" src="/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png" alt="$ A $" />) <img class="teximage" src="/files/tex/2326592ad6b621142c337b6acc2a4b724ca723f4.png" alt="$$\left\langle f_1 \times^{\left( D \right)} f_2 \right\rangle x = \bigcup \left\{ \langle f_1\rangle X \times^{\mathsf{FCD}} \langle f_2\rangle X \hspace{1em} | \hspace{1em} X \in \mathrm{atoms}^{\mathfrak{A}} x \right\}.$$" /> (The join operation is taken on the lattice of filters with reversed order.) </div> <p>A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.</p> </tr></td> </table> </td> </tr> </table> Porton, Victor category theory general topology Topology http://openproblemgarden.org/op/a_conjecture_about_direct_product_of_funcoids#comment Thu, 26 Jul 2012 22:41:04 +0200 porton 37540 at http://openproblemgarden.org