{"id":7189,"date":"2023-09-20T19:43:22","date_gmt":"2023-09-20T17:43:22","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=7189"},"modified":"2023-09-20T19:43:54","modified_gmt":"2023-09-20T17:43:54","slug":"scotts-formula","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=7189","title":{"rendered":"Scott&#8217;s formula"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">There is a well-known formula in domain theory, which, given a monotonic map <em>f<\/em> from a basis <em>B<\/em> of a continuous poset <em>X<\/em> to a dcpo <em>Y<\/em>, produces the largest continuous map <em>f&#8217;<\/em> defined on the whole of <em>X<\/em> and below <em>f<\/em> on <em>B<\/em>.  This is <em>Scott&#8217;s formula<\/em>: <em>f&#8217;<\/em>(<em>x<\/em>) \u225d sup<sub><em>b<\/em> \u2208 <em>B<\/em>, <em>b<\/em> \u226a <em>x<\/em><\/sub> <em>f<\/em>(<em>b<\/em>).  Generalizing this to the setting where <em>X<\/em> is a c-space, I will explain when exactly <em>f&#8217;<\/em> coincides with <em>f<\/em> on <em>B<\/em>, hence when <em>f&#8217;<\/em> is a continuous extension of <em>f<\/em>.  Also, using a motivation from the theory of continuous valuations, I will describe conditions under which any algebraic laws satisfied by <em>f<\/em> transfer to <em>f&#8217;<\/em>.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=7144\" data-type=\"page\" data-id=\"7144\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>There is a well-known formula in domain theory, which, given a monotonic map f from a basis B of a continuous poset X to a dcpo Y, produces the largest continuous map f&#8217; defined on the whole of X and &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=7189\">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":[],"class_list":["post-7189","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/7189","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=7189"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/7189\/revisions"}],"predecessor-version":[{"id":7194,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/7189\/revisions\/7194"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7189"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7189"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}