https://www.openproblemgarden.org/op/do_filters_complementive_to_a_given_filter_form_a_complete_lattice Comments for "Do filters complementive to a given filter form a complete lattice?" en https://www.openproblemgarden.org/op/do_filters_complementive_to_a_given_filter_form_a_complete_lattice <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/porton_victor">Porton</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/unsorted">Unsorted</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>Let <img class="teximage" src="/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png" alt="$ U $" /> is a set. A filter (on <img class="teximage" src="/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png" alt="$ U $" />) <img class="teximage" src="/files/tex/b85926cf6e40aee694107b903bd7a2aaaa15bd2e.png" alt="$ \mathcal{F} $" /> is by definition a non-empty set of subsets of <img class="teximage" src="/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png" alt="$ U $" /> such that <img class="teximage" src="/files/tex/fa02363ab1114f4e22d877c8d236fe6eb56f8716.png" alt="$ A,B\in\mathcal{F} \Leftrightarrow A\cap B\in\mathcal{F} $" />. Note that unlike some other authors I do not require <img class="teximage" src="/files/tex/3e22961193a920629d6d3499cd73ff367d15387e.png" alt="$ \varnothing\notin\mathcal{F} $" />. I will denote <img class="teximage" src="/files/tex/074c1d0ee51de94591bc1fc56e893ff8f65f2568.png" alt="$ \mathscr{F} $" /> the lattice of all filters (on <img class="teximage" src="/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png" alt="$ U $" />) ordered by set inclusion.</p> <p>Let <img class="teximage" src="/files/tex/9958394f8f5cc59ef1b994bf56702e1967db5bc1.png" alt="$ \mathcal{A}\in\mathscr{F} $" /> is some (fixed) filter. Let <img class="teximage" src="/files/tex/45f752546cc0a78b816dbfa6a89a7bd2fa40e21d.png" alt="$ D=\{\mathcal{X}\in\mathscr{F} | \mathcal{X}\supseteq \mathcal{A}\} $" />. Obviously <img class="teximage" src="/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png" alt="$ D $" /> is a bounded lattice.</p> <p>I will call complementive such filters <img class="teximage" src="/files/tex/bdd1d8154093a87de6fd3370fcfbdc06e6a04b43.png" alt="$ \mathcal{C} $" /> that: <ol> <li><img class="teximage" src="/files/tex/c5fb95ddc99c60ed1c7fc7db17bdaff712e72e94.png" alt="$ \mathcal{C}\in D $" />;</li> <li><img class="teximage" src="/files/tex/bdd1d8154093a87de6fd3370fcfbdc06e6a04b43.png" alt="$ \mathcal{C} $" /> is a complemented element of the lattice <img class="teximage" src="/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png" alt="$ D $" />.</li> </ol> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; The set of complementive filters ordered by inclusion is a complete lattice. </div> </tr></td> </table> </td> </tr> </table> Porton, Victor complete lattice filter Unsorted https://www.openproblemgarden.org/op/do_filters_complementive_to_a_given_filter_form_a_complete_lattice#comment Fri, 31 Jul 2009 15:10:44 +0200 porton 37005 at https://www.openproblemgarden.org