http://openproblemgarden.org/op/chain_meet_closed_sets Comments for "Chain-meet-closed sets" en http://openproblemgarden.org/op/chain_meet_closed_sets <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/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" /> is a complete lattice. I will call a <em>filter base</em> a nonempty subset <img class="teximage" src="/files/tex/79f55d2e1d83a7726c807a70cbe756713b0437b6.png" alt="$ T $" /> of <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" /> such that <img class="teximage" src="/files/tex/c85e9eaacec285000b660c7c9bc535f52c12b83c.png" alt="$ \forall a,b\in T\exists c\in T: (c\le a\wedge c\le b) $" />.</p> <div class="envsimple"><b>Definition</b>&nbsp;&nbsp; A subset <img class="teximage" src="/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png" alt="$ S $" /> of a complete lattice <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" /> is <em>chain-meet-closed</em> iff for every non-empty <a href="http://en.wikipedia.org/wiki/Total_order#Chains">chain</a> <img class="teximage" src="/files/tex/57e116e15e721bcdb86104a2c28f1cdfb0e6df9e.png" alt="$ T\in\mathscr{P}S $" /> we have <img class="teximage" src="/files/tex/8f696ee668ef8dd7ea6c8a64b9669645285fc295.png" alt="$ \bigcap T\in S $" />. </div> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; A subset <img class="teximage" src="/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png" alt="$ S $" /> of a complete lattice <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" /> is chain-meet-closed iff for every filter base <img class="teximage" src="/files/tex/57e116e15e721bcdb86104a2c28f1cdfb0e6df9e.png" alt="$ T\in\mathscr{P}S $" /> we have <img class="teximage" src="/files/tex/8f696ee668ef8dd7ea6c8a64b9669645285fc295.png" alt="$ \bigcap T\in S $" />. </div> </tr></td> </table> </td> </tr> </table> Porton, Victor chain complete lattice filter bases filters linear order total order Unsorted http://openproblemgarden.org/op/chain_meet_closed_sets#comment Sat, 12 Dec 2009 20:22:52 +0100 porton 37171 at http://openproblemgarden.org