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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.00440 |
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| _version_ | 1866908342037774336 |
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| author | Bhardwaj, Neer Moonen, Frodo |
| author_facet | Bhardwaj, Neer Moonen, Frodo |
| contents | Let $G$ be a proper subgroup of $\mathbb{Q}$ and $S_G$ be the set of primes $p$ for which $G$ is $p$-divisible. We show that the model-theoretic Grothendieck ring of the ordered abelian group $(G;+,<)$ is a quotient of $(\mathbb{Z}/q\mathbb{Z})[T]/(T+T^2)$, where $q$ is the largest odd integer that divides $p-1$ for all $p \notin S_G$.
This implies that the Grothendieck ring of $(G;+,<)$ is trivial in various salient cases, for example when $S_G$ is finite, or when $S_G$ does not contain some prime of the form $2^n+1$, $n\in \mathbb{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00440 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Grothendieck rings of ordered subgroups of $\mathbb{Q}$ Bhardwaj, Neer Moonen, Frodo Rings and Algebras Logic 3C07, 03C64, 06F20 Let $G$ be a proper subgroup of $\mathbb{Q}$ and $S_G$ be the set of primes $p$ for which $G$ is $p$-divisible. We show that the model-theoretic Grothendieck ring of the ordered abelian group $(G;+,<)$ is a quotient of $(\mathbb{Z}/q\mathbb{Z})[T]/(T+T^2)$, where $q$ is the largest odd integer that divides $p-1$ for all $p \notin S_G$. This implies that the Grothendieck ring of $(G;+,<)$ is trivial in various salient cases, for example when $S_G$ is finite, or when $S_G$ does not contain some prime of the form $2^n+1$, $n\in \mathbb{N}$. |
| title | Grothendieck rings of ordered subgroups of $\mathbb{Q}$ |
| topic | Rings and Algebras Logic 3C07, 03C64, 06F20 |
| url | https://arxiv.org/abs/2503.00440 |