{"id":7138,"date":"2023-08-20T09:42:13","date_gmt":"2023-08-20T07:42:13","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=7138"},"modified":"2023-08-20T09:44:40","modified_gmt":"2023-08-20T07:44:40","slug":"the-fundamental-theorem-of-compact-semilattices","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=7138","title":{"rendered":"The fundamental theorem of compact semilattices"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Bounded-complete domains, or bc-domains, are an amazingly rich kind of continuous domains.  They form a Cartesian-closed category, and they are the densely injective topological spaces, among other properties.  One characterization of bc-domains which I have not included in the book is that they are related in a very precise sense to Lawson semilattices, namely to compact semilattices with small semilattices.  This is the <em>fundamental theorem of compact semilattices<\/em>, which I will (re)prove by relying a lot on the theory of compact pospaces and stably compact spaces: see the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=7078\" data-type=\"page\" data-id=\"7078\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bounded-complete domains, or bc-domains, are an amazingly rich kind of continuous domains. They form a Cartesian-closed category, and they are the densely injective topological spaces, among other properties. One characterization of bc-domains which I have not included in the book &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=7138\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","footnotes":""},"categories":[1],"tags":[77,76],"class_list":["post-7138","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-compact-pospace","tag-semilattice"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/7138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7138"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/7138\/revisions"}],"predecessor-version":[{"id":7139,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/7138\/revisions\/7139"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}