Topological representations for frame-valued domains via $L$-sobriety
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910560162938880 |
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| author | Wu, Guojun Yao, Wei Li, Qingguo |
| author_facet | Wu, Guojun Yao, Wei Li, Qingguo |
| contents | 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 $(P, σ_{L}(P))$ with continuous maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological representations for frame-valued domains via $L$-sobriety Wu, Guojun Yao, Wei Li, Qingguo General Topology 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 $(P, σ_{L}(P))$ with continuous maps. |
| title | Topological representations for frame-valued domains via $L$-sobriety |
| topic | General Topology |
| url | https://arxiv.org/abs/2406.13595 |