Guojun WuSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Wei YaoSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Qingguo LiSchool of Mathematics, Hunan University
{"title":"Topological representations for frame-valued domains via $L$-sobriety","authors":"Guojun WuSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Wei YaoSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Qingguo LiSchool of Mathematics, Hunan University","doi":"arxiv-2406.13595","DOIUrl":null,"url":null,"abstract":"With a frame $L$ as the truth value table, we study the topological\nrepresentations for frame-valued domains. We introduce the notions of locally\nsuper-compact $L$-topological space and strong locally super-compact\n$L$-topological space. Using these concepts, continuous $L$-dcpos and algebraic\n$L$-dcpos are successfully represented via $L$-sobriety. By means of Scott\n$L$-topology and specialization $L$-order, we establish a categorical\nisomorphism between the category of the continuous (resp., algebraic) $L$-dcpos\nwith Scott continuous maps and that of the locally super-compact (resp., strong\nlocally super-compact) $L$-sober spaces with continuous maps. As an\napplication, for a continuous $L$-poset $P$, we obtain a categorical\nisomorphism between the category of directed completions of $P$ with Scott\ncontinuous maps and that of the $L$-sobrifications of $(P, \\sigma_{L}(P))$ with\ncontinuous maps.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"208 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - General Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2406.13595","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
With a frame $L$ as the truth value table, we study the topological
representations for frame-valued domains. We introduce the notions of locally
super-compact $L$-topological space and strong locally super-compact
$L$-topological space. Using these concepts, continuous $L$-dcpos and algebraic
$L$-dcpos are successfully represented via $L$-sobriety. By means of Scott
$L$-topology and specialization $L$-order, we establish a categorical
isomorphism between the category of the continuous (resp., algebraic) $L$-dcpos
with Scott continuous maps and that of the locally super-compact (resp., strong
locally super-compact) $L$-sober spaces with continuous maps. As an
application, for a continuous $L$-poset $P$, we obtain a categorical
isomorphism between the category of directed completions of $P$ with Scott
continuous maps and that of the $L$-sobrifications of $(P, \sigma_{L}(P))$ with
continuous maps.