http://openproblemgarden.org/op/a_construction_on_boolean_lattices_is_itself_a_boolean_lattice Comments for "A construction on boolean lattices is itself a boolean lattice?" en http://openproblemgarden.org/op/a_construction_on_boolean_lattices_is_itself_a_boolean_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>Let <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" /> and <img class="teximage" src="/files/tex/480eddb2f10987e813f894e1a635408d9782fb53.png" alt="$ \mathfrak{B} $" /> be (fixed) boolean lattices (with lattice operations denoted <img class="teximage" src="/files/tex/8b9a17030b1283e32dd72bd0a22109dec122655d.png" alt="$ \sqcup $" /> and <img class="teximage" src="/files/tex/3c04961b52bb2bd4a0d12a73b6b31ebf8a5d2813.png" alt="$ \sqcap $" />, bottom element <img class="teximage" src="/files/tex/fe876c19fffd84ea4caee40aa24c4d1234e18152.png" alt="$ \bot $" /> and top element <img class="teximage" src="/files/tex/3b4adf65153bef1c4fa3f763de0dbc319f66037f.png" alt="$ \top $" />).</p> <p>I call a <em>boolean funcoid</em> a pair <img class="teximage" src="/files/tex/66c6f81b751b25290ec6d2a361aed65c925efc7a.png" alt="$ (\alpha;\beta) $" /> of functions <img class="teximage" src="/files/tex/ce042c0c0a7ce66e67f2abb27046d47e5bb199d8.png" alt="$ \alpha:\mathfrak{A}\rightarrow\mathfrak{B} $" />, <img class="teximage" src="/files/tex/012c7084913f2fb5e3d0ea6c7bf0d35fe3edaed7.png" alt="$ \beta:\mathfrak{B}\rightarrow\mathfrak{A} $" /> such that (for every <img class="teximage" src="/files/tex/eb4cd59276f7b1d60a23538239b12131f99a9cd3.png" alt="$ X\in\mathfrak{A} $" />, <img class="teximage" src="/files/tex/d0d417874a12bbd70529f745c1fc6005ed62eb09.png" alt="$ Y\in\mathfrak{B} $" />) <img class="teximage" src="/files/tex/10cfbbfaeee162a8c7ffe2a3f22e316716d7dae6.png" alt="$$Y\sqcap^{\mathfrak{B}}\alpha(X)\ne\bot^{\mathfrak{B}} \Leftrightarrow X\sqcap^{\mathfrak{A}}\beta(Y)\ne\bot^{\mathfrak{A}}.$$" /></p> <p>(Boolean funcoids are a special case of pointfree funcoids as defined in my <a href="http://www.mathematics21.org/algebraic-general-topology.html">free ebook</a>.)</p> <p>Order boolean funcoids by the formula <img class="teximage" src="/files/tex/b8370f91ae4537c4a62137ad101d6f7ef32f92de.png" alt="$$(\alpha_0;\beta_0)\le (\alpha_1;\beta_1) \Leftrightarrow \forall X\in\mathfrak{A}: \alpha_0(X)\le\alpha_1(X) \land \forall Y\in\mathfrak{B}: \beta_0(Y)\le\beta_1(Y).$$" /></p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; The set of boolean funcoids with above defined order is itself a boolean lattice. </div> <p>If this conjecture does not hold in general, does it hold for: a. atomic boolean lattices? b. atomistic boolean lattices? c. complete boolean lattices?</p> <p>For the special case when <img class="teximage" src="/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png" alt="$ \mathfrak{A} $" /> and <img class="teximage" src="/files/tex/480eddb2f10987e813f894e1a635408d9782fb53.png" alt="$ \mathfrak{B} $" /> are complete atomic boolean lattices, the conjecture easily follows from <a href="http://math.stackexchange.com/questions/1370745/two-adjunct-quasi-inverse-functions/1370753#1370753">this math.SE answer</a>.</p> </tr></td> </table> </td> </tr> </table> Porton, Victor boolean algebra boolean lattice lattice Algebra http://openproblemgarden.org/op/a_construction_on_boolean_lattices_is_itself_a_boolean_lattice#comment Mon, 12 Oct 2015 21:24:34 +0200 porton 60007 at http://openproblemgarden.org