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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2512.18327 |
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| _version_ | 1866917160053374976 |
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| author | Chini, Leon |
| author_facet | Chini, Leon |
| contents | This paper further studies the model companion of an endomorphism acting on a vector space, possibly with extra structure. Given a theory $T$ that $\varnothing$-defines an infinite $K$-vector space $\mathbb{V}$ in every model, we set $T_θ:= T \cup \{\text{``$θ$ defines a $K$-endomorphism of $\mathbb{V}$"}\}$. We previously defined a family $\{T^C_θ: C \in \mathcal{C}\}$ of extensions of $T_θ$ which parameterizes all consistent extensions of the form $$
T_θ\cup \left\{\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(ρ_{j, k, l}[θ]) = \sum\nolimits_{k}\bigcap\nolimits_{l} \operatorname{Ker}(η_{j, k, l}[θ]) : j \in \mathcal{J}\right\}, $$ where all sums and intersections are finite, and all the $ρ[θ]$'s and $η[θ]$'s are polynomials over $K$ with $θ$ plugged in. Notice that properties such as $θ^2 - 2\operatorname{Id} = 0$ or ``$ρ[θ]$ is injective for every $ρ\in K[X] \setminus \{0\}$" can be expressed in such a manner. We also presented a sufficient condition which implies that every $T^C_θ$ has a model companion $Tθ^C$. Under this condition, we characterize all definable sets in $Tθ^C$ and use this to study the completions of $Tθ^C$, as well as the algebraic closure. If $T$ is o-minimal and extends $\operatorname{Th}(\mathbb{R}, <)$, we prove that $Tθ^C$ has o-minimal open core. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18327 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Model Theory of Generic Vector Space Endomorphisms II Chini, Leon Logic 03C10 (Primary) 03C60 (Secondary) This paper further studies the model companion of an endomorphism acting on a vector space, possibly with extra structure. Given a theory $T$ that $\varnothing$-defines an infinite $K$-vector space $\mathbb{V}$ in every model, we set $T_θ:= T \cup \{\text{``$θ$ defines a $K$-endomorphism of $\mathbb{V}$"}\}$. We previously defined a family $\{T^C_θ: C \in \mathcal{C}\}$ of extensions of $T_θ$ which parameterizes all consistent extensions of the form $$ T_θ\cup \left\{\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(ρ_{j, k, l}[θ]) = \sum\nolimits_{k}\bigcap\nolimits_{l} \operatorname{Ker}(η_{j, k, l}[θ]) : j \in \mathcal{J}\right\}, $$ where all sums and intersections are finite, and all the $ρ[θ]$'s and $η[θ]$'s are polynomials over $K$ with $θ$ plugged in. Notice that properties such as $θ^2 - 2\operatorname{Id} = 0$ or ``$ρ[θ]$ is injective for every $ρ\in K[X] \setminus \{0\}$" can be expressed in such a manner. We also presented a sufficient condition which implies that every $T^C_θ$ has a model companion $Tθ^C$. Under this condition, we characterize all definable sets in $Tθ^C$ and use this to study the completions of $Tθ^C$, as well as the algebraic closure. If $T$ is o-minimal and extends $\operatorname{Th}(\mathbb{R}, <)$, we prove that $Tθ^C$ has o-minimal open core. |
| title | Model Theory of Generic Vector Space Endomorphisms II |
| topic | Logic 03C10 (Primary) 03C60 (Secondary) |
| url | https://arxiv.org/abs/2512.18327 |