Topological representations for frame-valued domains via $L$-sobriety

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
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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.
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通过 $L$-sobriety 实现框架值域的拓扑表征
以帧$L$为真值表,我们研究帧值域的拓扑表示。我们引入了局部超紧密$L$拓扑空间和强局部超紧密$L$拓扑空间的概念。利用这些概念,连续$L$-dcpos和代数$L$-dcpos可以通过$L$-sobriety成功地表示出来。通过斯科特$L$拓扑学和特化$L$阶,我们在具有斯科特连续映射的连续(或代数)$L$-dcpos范畴和具有连续映射的局部超紧密(或强局部超紧密)$L$-清醒空间范畴之间建立了一种分类同构关系。作为应用,对于连续的$L$-poset $P$,我们得到了具有斯科特连续映射的$P$的有向补全类别与具有连续映射的$(P, \sigma_{L}(P))$的$L$-sobrifications类别之间的分类同构。
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