http://openproblemgarden.org/op/a_diagram_about_funcoids_and_reloids Comments for "A diagram about funcoids and reloids" en http://openproblemgarden.org/op/a_diagram_about_funcoids_and_reloids#comment-79539 <p>It seems that the node "other" is not <img class="teximage" src="/files/tex/4a511edece8921fab6426695d3451efc024273a5.png" alt="$ \lambda f\in\mathsf{FCD}: \top $" />.</p> <p>I conjecture <img class="teximage" src="/files/tex/80dcbb1586646ba53f53f2eeffb9216a9fbf7980.png" alt="$ \langle \Phi_{\ast} (\mathsf{RLD})_{\operatorname{out}} \rangle f = (\mathsf{FCD}) f $" /> where <img class="teximage" src="/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png" alt="$ f $" /> is the reloid defined by the cofinite filter on <img class="teximage" src="/files/tex/0a682126d12892c7763052125365ba4bbb0ec49b.png" alt="$ A \times B $" /> and thus <img class="teximage" src="/files/tex/b31f3f0a1acdc75f6cc233fee46c5ef6a32a6941.png" alt="$ \langle (\mathsf{FCD}) f \rangle \{ x \} = \bot $" /> for all singletons <img class="teximage" src="/files/tex/3a418097d683651fdb01812be70bd8c8aaf0c1c5.png" alt="$ \{ x \} $" /> and <img class="teximage" src="/files/tex/599fb01c219d59815efc36479674e1f7f748cd55.png" alt="$ \langle (\mathsf{FCD}) f \rangle p = \top $" /> for every nontrivial atomic filter <img class="teximage" src="/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png" alt="$ p $" />.</p> <p>This is my very recent thoughts and yet needs to be checked.</p> <p>-- Victor Porton - http://www.mathematics21.org</p> Mon, 28 Nov 2016 23:24:38 +0100 porton comment 79539 at http://openproblemgarden.org http://openproblemgarden.org/op/a_diagram_about_funcoids_and_reloids#comment-79516 <p>My diagram was with an error. I have uploaded a corrected version of the diagram.</p> <p>--<br /> Victor Porton - <a href="http://www.mathematics21.org" title="http://www.mathematics21.org">http://www.mathematics21.org</a></p> Sat, 26 Nov 2016 19:13:28 +0100 porton comment 79516 at http://openproblemgarden.org http://openproblemgarden.org/op/a_diagram_about_funcoids_and_reloids <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/porton_victor">Porton</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/topology">Topology</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>Define for posets with order <img class="teximage" src="/files/tex/b459f2ec7c3664b435b4c1e91359e4d32338d557.png" alt="$ \sqsubseteq $" />: <ol> <li><img class="teximage" src="/files/tex/534febaf8df5ac08ab098b143b37a9333cb8cef2.png" alt="$ \Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \} $" />; <li><img class="teximage" src="/files/tex/7fa2f9484869d9b89bb57e2d761bc8dd7e449b40.png" alt="$ \Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigcap \{ x \in \mathfrak{B} \mid f x \sqsupseteq b \} $" />. </ol> <p>Note that the above is a generalization of monotone Galois connections (with <img class="teximage" src="/files/tex/913d4907dab02a1dd155d8911760693cc1da5b79.png" alt="$ \max $" /> and <img class="teximage" src="/files/tex/2ceca20a9b83c7889bae4e31b22a10597d835b2f.png" alt="$ \min $" /> replaced with suprema and infima).</p> <p>Then we have the following diagram:</p> <p><img src="files/diagram2.png" align="center"></p> <p>What is at the node "other" in the diagram is unknown.</p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; "Other" is <img class="teximage" src="/files/tex/4a511edece8921fab6426695d3451efc024273a5.png" alt="$ \lambda f\in\mathsf{FCD}: \top $" />. </div> <div class="envtheorem"><b>Question</b>&nbsp;&nbsp; What repeated applying of <img class="teximage" src="/files/tex/c26f0d43856d263f335939666a99f483ffd09da8.png" alt="$ \Phi_{\ast} $" /> and <img class="teximage" src="/files/tex/cf6796f6de9023eb4c0ae3e69b8900a93e53fc6b.png" alt="$ \Phi^{\ast} $" /> to "other" leads to? Particularly, does repeated applying <img class="teximage" src="/files/tex/c26f0d43856d263f335939666a99f483ffd09da8.png" alt="$ \Phi_{\ast} $" /> and/or <img class="teximage" src="/files/tex/cf6796f6de9023eb4c0ae3e69b8900a93e53fc6b.png" alt="$ \Phi^{\ast} $" /> to the node "other" lead to finite or infinite sets? </div> </tr></td> </table> </td> </tr> </table> Porton, Victor Galois connections Topology http://openproblemgarden.org/op/a_diagram_about_funcoids_and_reloids#comment Sat, 26 Nov 2016 17:19:32 +0100 porton 60024 at http://openproblemgarden.org