http://www.openproblemgarden.org/op/cross_composition_product_of_reloids_is_a_quasi_cartesian_function Comments for "Cross-composition product of reloids is a quasi-cartesian function" en http://www.openproblemgarden.org/op/cross_composition_product_of_reloids_is_a_quasi_cartesian_function <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; Cross-composition product (for small indexed families of reloids) is a quasi-cartesian function (with injective aggregation) from the quasi-cartesian situation <img class="teximage" src="/files/tex/fd17cb814da4239e3c1e5b6282e71bd664f2724b.png" alt="$ \mathfrak{S}_0 $" /> of reloids to the quasi-cartesian situation <img class="teximage" src="/files/tex/63d285ad20884d993d07c83602cd3e7856596cfc.png" alt="$ \mathfrak{S}_1 $" /> of pointfree funcoids over posets with least elements. </div> <p>This conjecture is unsolved even for product of two multipliers.</p> <p>An obviously equivalent reformulation of this conjecture for the special case of two multipliers:</p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Provided that reloids <img class="teximage" src="/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png" alt="$ f $" /> and <img class="teximage" src="/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png" alt="$ g $" /> on some set <img class="teximage" src="/files/tex/8a60ffddf5f015b5658f273c0698af378fdebbde.png" alt="$ \mho $" /> are proper filters, we can restore the values of <img class="teximage" src="/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png" alt="$ f $" /> and <img class="teximage" src="/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png" alt="$ g $" /> knowing only <img class="teximage" src="/files/tex/8b75baa7837c3ec36a34b939706fbc0294c26323.png" alt="$ g\circ a\circ f^{-1} $" /> for every reloid <img class="teximage" src="/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png" alt="$ a $" /> on <img class="teximage" src="/files/tex/8a60ffddf5f015b5658f273c0698af378fdebbde.png" alt="$ \mho $" />. </div> <p>Reloids are defined simply as filters on a Cartesian product of two sets. The reverse reloid <img class="teximage" src="/files/tex/f9e76bad77ee7b47bb7937393ee8bde41bdf733e.png" alt="$ f^{-1} $" /> of a reloids <img class="teximage" src="/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png" alt="$ f $" /> is defined by the formula: <img class="teximage" src="/files/tex/ca3d3ab508da31a762274c5e3452de169ef1ae73.png" alt="$ f^{-1} = \{F^{-1} \,|\, F\in f \} $" />. Composition <img class="teximage" src="/files/tex/38baf2e90253c78467bd8e0cf322258244aab4f9.png" alt="$ g\circ f $" /> of reloids <img class="teximage" src="/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png" alt="$ f $" /> and <img class="teximage" src="/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png" alt="$ g $" /> is defined as the reloid whose base is <img class="teximage" src="/files/tex/96ad61c3a80d5b3cea7e3d3e9f1493f8afdb4646.png" alt="$$\{ G\circ F \,|\, F\in f, G\in g\}.$$" /></p> </tr></td> </table> </td> </tr> </table> Porton, Victor cross-composition product quasi-cartesian function Topology http://www.openproblemgarden.org/op/cross_composition_product_of_reloids_is_a_quasi_cartesian_function#comment Thu, 05 Jul 2012 22:53:48 +0200 porton 37486 at http://www.openproblemgarden.org