Growing Spines: Ad Infinitum et Ad Infinitesimalia
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909943653728256 |
|---|---|
| author | Boissonneau, Blaise De Mase, Anna Jahnke, Franziska Touchard, Pierre |
| author_facet | Boissonneau, Blaise De Mase, Anna Jahnke, Franziska Touchard, Pierre |
| contents | We prove that for every ordered abelian group $G$ there exists a non-trivial ordered abelian group $H$ such that $G\preccurlyeq H\oplus G$ with the lexicographic order, and give a first-order characterization of ordered abelian group $G$ such that $G\preccurlyeq G\oplus H$ for some non-trivial $H$. We apply this to characterize which ordered abelian groups (respectively fields) ensure that any henselian valuation with said value group (respectively residue field) is definable in the language of rings. This answers a question of Krapp, Kuhlmann, and Link. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04932 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Growing Spines: Ad Infinitum et Ad Infinitesimalia Boissonneau, Blaise De Mase, Anna Jahnke, Franziska Touchard, Pierre Logic 03C60, 03C64, 06F20 (Primary), 12J20, 12L12 (Secondary) We prove that for every ordered abelian group $G$ there exists a non-trivial ordered abelian group $H$ such that $G\preccurlyeq H\oplus G$ with the lexicographic order, and give a first-order characterization of ordered abelian group $G$ such that $G\preccurlyeq G\oplus H$ for some non-trivial $H$. We apply this to characterize which ordered abelian groups (respectively fields) ensure that any henselian valuation with said value group (respectively residue field) is definable in the language of rings. This answers a question of Krapp, Kuhlmann, and Link. |
| title | Growing Spines: Ad Infinitum et Ad Infinitesimalia |
| topic | Logic 03C60, 03C64, 06F20 (Primary), 12J20, 12L12 (Secondary) |
| url | https://arxiv.org/abs/2512.04932 |