Semilattice base hierarchy for frames and its topological ramifications
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911850092822528 |
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| author | Bezhanishvili, G. Dashiell, F. Razafindrakoto, A. Walters-Wayland, J. |
| author_facet | Bezhanishvili, G. Dashiell, F. Razafindrakoto, A. Walters-Wayland, J. |
| contents | We develop a hierarchy of semilattice bases (S-bases) for frames. For a given (unbounded) meet-semilattice $A$, we analyze the interval in the coframe of sublocales of the frame of downsets of $A$ formed by all frames with the S-base $A$. We give an explicit description of the nuclei associated with these sublocales. We study various degrees of completeness of $A$, which generalize the concepts of extremally disconnected and basically disconnected frames. We also introduce the concepts of D-bases and L-bases, as well as their bounded counterparts, and show how our results specialize and sharpen in these cases. Classic examples that are covered by our approach include zero-dimensional, completely regular, and coherent frames, allowing us to provide a new perspective on these well-studied classes of frames, as well as their spatial counterparts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_01627 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Semilattice base hierarchy for frames and its topological ramifications Bezhanishvili, G. Dashiell, F. Razafindrakoto, A. Walters-Wayland, J. General Topology We develop a hierarchy of semilattice bases (S-bases) for frames. For a given (unbounded) meet-semilattice $A$, we analyze the interval in the coframe of sublocales of the frame of downsets of $A$ formed by all frames with the S-base $A$. We give an explicit description of the nuclei associated with these sublocales. We study various degrees of completeness of $A$, which generalize the concepts of extremally disconnected and basically disconnected frames. We also introduce the concepts of D-bases and L-bases, as well as their bounded counterparts, and show how our results specialize and sharpen in these cases. Classic examples that are covered by our approach include zero-dimensional, completely regular, and coherent frames, allowing us to provide a new perspective on these well-studied classes of frames, as well as their spatial counterparts. |
| title | Semilattice base hierarchy for frames and its topological ramifications |
| topic | General Topology |
| url | https://arxiv.org/abs/2308.01627 |