Sober $L$-convex spaces and $L$-join-semilattices
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929461123874816 |
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| author | Wu, Guojun Yao, Wei |
| author_facet | Wu, Guojun Yao, Wei |
| contents | With a complete residuated lattice $L$ as the truth value table, we extend the definition of sobriety of classical convex spaces to the framework of $L$-convex spaces. We provide a specific construction for the sobrification of an $L$-convex space, demonstrating that the full subcategory of sober $L$-convex spaces is reflective in the category of $L$-convex spaces with convexity-preserving mappings. Additionally, we introduce the concept of Scott $L$-convex structures on $L$-ordered sets. As an application of this type of sobriety, we obtain a characterization for the $L$-join-semilattice completion of an $L$-ordered set: an $L$-ordered set $Q$ is an $L$-join-semilattice completion of an $L$-ordered set $P$ if and only if the Scott $L$-convex space $(Q, σ^{\ast}(Q))$ is a sobrification of the Scott $L$-convex space $(P, σ^{\ast}(P))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08520 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sober $L$-convex spaces and $L$-join-semilattices Wu, Guojun Yao, Wei General Topology Category Theory With a complete residuated lattice $L$ as the truth value table, we extend the definition of sobriety of classical convex spaces to the framework of $L$-convex spaces. We provide a specific construction for the sobrification of an $L$-convex space, demonstrating that the full subcategory of sober $L$-convex spaces is reflective in the category of $L$-convex spaces with convexity-preserving mappings. Additionally, we introduce the concept of Scott $L$-convex structures on $L$-ordered sets. As an application of this type of sobriety, we obtain a characterization for the $L$-join-semilattice completion of an $L$-ordered set: an $L$-ordered set $Q$ is an $L$-join-semilattice completion of an $L$-ordered set $P$ if and only if the Scott $L$-convex space $(Q, σ^{\ast}(Q))$ is a sobrification of the Scott $L$-convex space $(P, σ^{\ast}(P))$. |
| title | Sober $L$-convex spaces and $L$-join-semilattices |
| topic | General Topology Category Theory |
| url | https://arxiv.org/abs/2408.08520 |