Definable ranks
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | |
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| _version_ | 1866909986175582208 |
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| author | Krapp, Lothar Sebastian Kuhlmann, Salma Vogel, Lasse |
| author_facet | Krapp, Lothar Sebastian Kuhlmann, Salma Vogel, Lasse |
| contents | We introduce the notion of the definable rank of an ordered field, ordered abelian group and ordered set, respectively. We study the relation between the definable rank of an ordered field and the definable rank of the value group of its natural valuation. Similarly, we compare the definable rank of an ordered abelian group to that of its value set with respect to the natural valuation. We describe the definable rank on the group-level by characterizing the definable convex subgroups. We also give a detailed comparison of field- and group-level, in particular for ordered fields with henselian natural valuation. We investigate definability of final segments in ordered sets and introduce definable condensation as a tool for further study. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00443 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Definable ranks Krapp, Lothar Sebastian Kuhlmann, Salma Vogel, Lasse Logic Commutative Algebra 03C64, 03C40 (Primary) 12J15, 12L12, 12J10, 06F20, 06A05 (Secondary) We introduce the notion of the definable rank of an ordered field, ordered abelian group and ordered set, respectively. We study the relation between the definable rank of an ordered field and the definable rank of the value group of its natural valuation. Similarly, we compare the definable rank of an ordered abelian group to that of its value set with respect to the natural valuation. We describe the definable rank on the group-level by characterizing the definable convex subgroups. We also give a detailed comparison of field- and group-level, in particular for ordered fields with henselian natural valuation. We investigate definability of final segments in ordered sets and introduce definable condensation as a tool for further study. |
| title | Definable ranks |
| topic | Logic Commutative Algebra 03C64, 03C40 (Primary) 12J15, 12L12, 12J10, 06F20, 06A05 (Secondary) |
| url | https://arxiv.org/abs/2506.00443 |