http://openproblemgarden.org/op/non_separable_center_of_a_lattice Comments for "Non-separable center of a lattice" en http://openproblemgarden.org/op/non_separable_center_of_a_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/algebra">Algebra</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>I will call center <img class="teximage" src="/files/tex/630c6ee247415899826b8dc922897abd8fbf219a.png" alt="$ Z(\mathfrak{A}) $" /> of a bounded distributive lattice <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" /> the sublattice of all complemented elements of <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" />.</p> <p>I will call a bounded distributive lattice <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" /> a lattice with separable center when <img class="teximage" src="/files/tex/5e1ee582ceea72350004e494f1d1a819b094ac3f.png" alt="$$\forall x,y\in\mathfrak{A}: (x\cap y=0\Rightarrow \exists X\in Z(\mathfrak{A}):(x\subseteq X\wedge X\cap y = 0)) .$$" /></p> <p>Equivalently a bounded distributive lattice with separable center is such a bounded distributive lattice <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" /> that <img class="teximage" src="/files/tex/70b6f05d90ac57db42a80a05c36a920e626d34c8.png" alt="$$\forall x,y\in\mathfrak{A}:(x\cap y=0\Rightarrow\exists X,Y\in Z(\mathfrak{A}):(x\subseteq X\wedge y\subseteq Y\wedge X\cap Y = 0)) .$$" /></p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; There exist bounded distributive lattices which are not with separable center. </div> </tr></td> </table> </td> </tr> </table> Porton, Victor lattice Algebra http://openproblemgarden.org/op/non_separable_center_of_a_lattice#comment Thu, 11 Sep 2008 19:58:25 +0200 porton 1849 at http://openproblemgarden.org