http://openproblemgarden.org/op/what_are_hyperfuncoids_isomorphic_to Comments for "What are hyperfuncoids isomorphic to?" en http://openproblemgarden.org/op/what_are_hyperfuncoids_isomorphic_to <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> <p>Let <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" /> be an indexed family of sets.</p> <p><em>Products</em> are <img class="teximage" src="/files/tex/bdab150894a9c199927d3ffbe7a69feed3981506.png" alt="$ \prod A $" /> for <img class="teximage" src="/files/tex/cfa13398aac3db0978948b07591484e5ed90aadc.png" alt="$ A \in \prod \mathfrak{A} $" />.</p> <p><em>Hyperfuncoids</em> are filters <img class="teximage" src="/files/tex/2d4cbeff4993cf10008cbe69e72409840d1b2201.png" alt="$ \mathfrak{F} \Gamma $" /> on the lattice <img class="teximage" src="/files/tex/9315cbad9b261f93c6d27530930a83c5cc705b0c.png" alt="$ \Gamma $" /> of all finite unions of products.</p> <div class="envtheorem"><b>Problem</b>&nbsp;&nbsp; Is <img class="teximage" src="/files/tex/3c8ed688fbcae181a7b030c7071347137615d338.png" alt="$ \bigcap^{\mathsf{\tmop{FCD}}} $" /> a bijection from hyperfuncoids <img class="teximage" src="/files/tex/2d4cbeff4993cf10008cbe69e72409840d1b2201.png" alt="$ \mathfrak{F} \Gamma $" /> to:<br /> <ol class="enumerate"> \item prestaroids on <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" />; \item staroids on <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" />; \item completary staroids on <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" />? </ol> <p> If yes, is <img class="teximage" src="/files/tex/183fe66c541f4606e43d646c2444eaa15764c9ba.png" alt="$ \operatorname{up}^{\Gamma} $" /> defining the inverse bijection? If not, characterize the image of the function <img class="teximage" src="/files/tex/3c8ed688fbcae181a7b030c7071347137615d338.png" alt="$ \bigcap^{\mathsf{\tmop{FCD}}} $" /> defined on <img class="teximage" src="/files/tex/2d4cbeff4993cf10008cbe69e72409840d1b2201.png" alt="$ \mathfrak{F} \Gamma $" />.</p> <p> Consider also the variant of this problem with the set <img class="teximage" src="/files/tex/9315cbad9b261f93c6d27530930a83c5cc705b0c.png" alt="$ \Gamma $" /> replaced with the set <img class="teximage" src="/files/tex/7910d5f60958e26e9306a9c315e1c61d83833f9f.png" alt="$ \Gamma^{\ast} $" /> of complements of elements of the set <img class="teximage" src="/files/tex/9315cbad9b261f93c6d27530930a83c5cc705b0c.png" alt="$ \Gamma $" />. </div> </tr></td> </table> </td> </tr> </table> Porton, Victor hyperfuncoids multidimensional Topology http://openproblemgarden.org/op/what_are_hyperfuncoids_isomorphic_to#comment Tue, 09 Dec 2014 21:27:27 +0100 porton 59973 at http://openproblemgarden.org