http://openproblemgarden.org/op/pseudodifference_of_filter_objects Comments for "Pseudodifference of filter objects" en http://openproblemgarden.org/op/pseudodifference_of_filter_objects#comment-7167 <p>In "pseudodifference of filter", the link for formal definition of filter objects does not work. Y V TYAGI</p> Sat, 19 May 2012 08:20:11 +0200 Anonymous comment 7167 at http://openproblemgarden.org http://openproblemgarden.org/op/pseudodifference_of_filter_objects <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 $" /> is a set. A filter <img class="teximage" src="/files/tex/b85926cf6e40aee694107b903bd7a2aaaa15bd2e.png" alt="$ \mathcal{F} $" /> (on <img class="teximage" src="/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png" alt="$ U $" />) is a non-empty set of subsets of <img class="teximage" src="/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png" alt="$ U $" /> such that <img class="teximage" src="/files/tex/aab30dca01377459880edd0634a7f4b69e796cca.png" alt="$ A, B \in \mathcal{F} \Leftrightarrow A \cap B \in \mathcal{F} $" />. Note that unlike some other authors I do not require <img class="teximage" src="/files/tex/bfc4450a4048c58c855ac21b3abce2574600079d.png" alt="$ \emptyset \notin \mathcal{F} $" />.</p> <p>I will call <em>the set of filter objects</em> the set of filters ordered reverse to set theoretic inclusion of filters, with principal filters equated to the corresponding sets. <a href="http://www.mathematics21.org/binaries/filters.pdf">See here for the formal definition of filter objects</a>. I will denote <img class="teximage" src="/files/tex/a9e778d2445bf30ef58154eec95345db9dd21fed.png" alt="$ (\operatorname{up} a) $" /> the filter corresponding to a filter object <img class="teximage" src="/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png" alt="$ a $" />. I will denote the set of filter objects (on <img class="teximage" src="/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png" alt="$ U $" />) as <img class="teximage" src="/files/tex/2f80186ad9cd0260fcf25f804b47e285791e195e.png" alt="$ \mathfrak{F} $" />.</p> <p>I will denote <img class="teximage" src="/files/tex/ebcf1b624c595ffb2e4a5cde53c1c36c77224b85.png" alt="$ (\operatorname{atoms} a) $" /> the set of atomic lattice elements under a given lattice element <img class="teximage" src="/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png" alt="$ a $" />. If <img class="teximage" src="/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png" alt="$ a $" /> is a filter object, then <img class="teximage" src="/files/tex/ebcf1b624c595ffb2e4a5cde53c1c36c77224b85.png" alt="$ (\operatorname{atoms} a) $" /> is essentially the set of ultrafilters over <img class="teximage" src="/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png" alt="$ a $" />.</p> <div class="envtheorem"><b>Problem</b>&nbsp;&nbsp; Which of the following expressions are pairwise equal for all <img class="teximage" src="/files/tex/b7eafe90dea74a3ae6d8c4fae0a6b0922fafa64f.png" alt="$ a, b \in \mathfrak{F} $" /> for each set <img class="teximage" src="/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png" alt="$ U $" />? (If some are not equal, provide counter-examples.)<br /> <ol class="enumerate"> \item <img class="teximage" src="/files/tex/11a1b7cf38a567f6603986ed2d03fb499cd030d3.png" alt="$ \bigcap^{\mathfrak{F}} \left\{ z \in \mathfrak{F} | a \subseteq b \cup^{\mathfrak{F}} z \right\} $" />;</p> <p> \item <img class="teximage" src="/files/tex/9ea6f8f408f5bb2c22f1a89b01e8e7f9b7f25ec0.png" alt="$ \bigcup^{\mathfrak{F}} \left\{ z \in \mathfrak{F} | z \subseteq a \wedge z \cap^{\mathfrak{F}} b = \emptyset \right\} $" />;</p> <p> \item <img class="teximage" src="/files/tex/5983ab3690f97fa117adae35a52475a4b3600fa9.png" alt="$ \bigcup^{\mathfrak{F}} (\operatorname{atoms} a \setminus \operatorname{atoms} b) $" />;</p> <p> \item <img class="teximage" src="/files/tex/52c54efb25ab0058d3da188be4adcf2f76e1e11a.png" alt="$ \bigcup^{\mathfrak{F}} \left\{ a \cap^{\mathfrak{F}} (U\setminus B) | B \in \operatorname{up} b \right\} $" />. </ol> </div> </tr></td> </table> </td> </tr> </table> Porton, Victor filters pseudodifference Unsorted http://openproblemgarden.org/op/pseudodifference_of_filter_objects#comment Tue, 24 Nov 2009 18:28:31 +0100 porton 37165 at http://openproblemgarden.org