{"id":563,"date":"2024-03-10T18:27:26","date_gmt":"2024-03-10T18:27:26","guid":{"rendered":"https:\/\/pri.tum.ac.ke\/?page_id=563"},"modified":"2024-03-10T18:27:27","modified_gmt":"2024-03-10T18:27:27","slug":"a-little-more-on-ideals-associated-with-sublocales","status":"publish","type":"page","link":"https:\/\/pri.tum.ac.ke\/?page_id=563","title":{"rendered":"A Little More On Ideals Associated With Sublocales"},"content":{"rendered":"\n<p>O. IGHEDO, G. W. KIVUNGA, AND D. N. STEPHEN<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Abstract<\/h3>\n\n\n\n<p>Let <em>R<\/em><em>L <\/em>denote the ring of real-valued continuous functions on a completely regular frame <em>L<\/em>, <em>\u03b2L<\/em>and <em>\u03bbL<\/em>denote the Stone-C<sup>\u02c7<\/sup>ech compactifica- tion of <em>L <\/em>and the Lindel\u00a8of coreflection of <em>L<\/em>, respectively. There is a natural way of associating with each sublocale of <em>\u03b2L<\/em>two ideals of <em>R<\/em><em>L<\/em>, motivated by a similar situation in <em>C<\/em>(<em>X<\/em>). This research augments the work of T. Dube and Stephen D.N. on mapping ideals to sublocales, where they associate with each sublocale of <em>\u03bbL<\/em>an ideal of <em>R<\/em><em>L<\/em>in a manner similar to one of the ways one does it for sublocales of <em>\u03b2L<\/em>. Two other coreflections; namely, the realcompact and the paracompact coreflections are considered.<\/p>\n\n\n\n<p>In the talk, I will show that <em><strong>M<\/strong><\/em>-ideals of <em>R<\/em><em>L <\/em>indexed by sublocales of <em>\u03b2L <\/em>are precisely the intersections of maximal ideals of <em>R<\/em><em>L<\/em>. An <em><strong>M<\/strong><\/em>-ideal of <em>R<\/em><em>L <\/em>is <em>grounded <\/em>in case it is of the form <em><strong>M<\/strong><\/em><sub><em>S<\/em><\/sub>for some sublocale <em>S <\/em>of <em>L<\/em>. A similar definition is given for an <em><strong>O<\/strong><\/em>-ideal of <em>R<\/em><em>L<\/em>. Time allowing, I will present charac- terizations of <em><strong>M<\/strong><\/em>-ideals of <em>R<\/em><em>L <\/em>indexed by spatial sublocales of <em>\u03b2L<\/em>, and <em><strong>O<\/strong><\/em>-ideals of <em>R<\/em><em>L<\/em>indexed by closed sublocales of <em>\u03b2L<\/em>in terms of grounded maximal ideals of <em>R<\/em><em>L<\/em>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>O. IGHEDO, G. W. KIVUNGA, AND D. N. STEPHEN Abstract Let RL denote the ring of real-valued continuous functions on a completely regular frame L, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":113,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-563","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/563","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=563"}],"version-history":[{"count":1,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/563\/revisions"}],"predecessor-version":[{"id":564,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/563\/revisions\/564"}],"up":[{"embeddable":true,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/113"}],"wp:attachment":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=563"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}