Characterizing $ (\mathcal{F}, \mathcal{G}) $-syndetic, $ (\mathcal{F}, \mathcal{G}) $-thick, and related notions of size using derived sets along ultrafilters
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914200945688576 |
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| author | Burns, Shea D. Davenport, Dennis Frankson, Shakuan Griffin, Conner Johnson Jr., John H. Kebe, Malick |
| author_facet | Burns, Shea D. Davenport, Dennis Frankson, Shakuan Griffin, Conner Johnson Jr., John H. Kebe, Malick |
| contents | We characterize relative notions of syndetic and thick sets using, what we call, "derived" sets along ultrafilters. Manipulations of derived sets is a characteristic feature of algebra in the Stone-Čech compactification and its applications. Combined with the existence of idempotents and structure of the smallest ideal in closed subsemigroups of the Stone-Čch compactification, our particular use of derived sets adapt and generalize methods recently used by Griffin arXiv:2311.09436 to characterize relative piecewise syndetic sets. As an application, we define an algebraically interesting subset of the Stone-Čech compactification and show, in some ways, it shares structural properties analogous to the smallest ideal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05579 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing $ (\mathcal{F}, \mathcal{G}) $-syndetic, $ (\mathcal{F}, \mathcal{G}) $-thick, and related notions of size using derived sets along ultrafilters Burns, Shea D. Davenport, Dennis Frankson, Shakuan Griffin, Conner Johnson Jr., John H. Kebe, Malick General Topology 54D35, 54D80 (Primary) 22A15 (Secondary) We characterize relative notions of syndetic and thick sets using, what we call, "derived" sets along ultrafilters. Manipulations of derived sets is a characteristic feature of algebra in the Stone-Čech compactification and its applications. Combined with the existence of idempotents and structure of the smallest ideal in closed subsemigroups of the Stone-Čch compactification, our particular use of derived sets adapt and generalize methods recently used by Griffin arXiv:2311.09436 to characterize relative piecewise syndetic sets. As an application, we define an algebraically interesting subset of the Stone-Čech compactification and show, in some ways, it shares structural properties analogous to the smallest ideal. |
| title | Characterizing $ (\mathcal{F}, \mathcal{G}) $-syndetic, $ (\mathcal{F}, \mathcal{G}) $-thick, and related notions of size using derived sets along ultrafilters |
| topic | General Topology 54D35, 54D80 (Primary) 22A15 (Secondary) |
| url | https://arxiv.org/abs/2503.05579 |