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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2502.13667 |
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| _version_ | 1866913888895762432 |
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| author | Chini, Leon |
| author_facet | Chini, Leon |
| contents | This paper deals with 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 then consider 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 $\operatorname{Ker}(θ^n) = \operatorname{Ker}(θ^{n+1})$ can be expressed in such a manner. We then parametrize the consistent extensions of this form by a family $\{T^C_θ: C \in \mathcal{C}\}$ and characterize the existentially closed models of each $T^C_θ$. We also present a sufficient criterion, which only depends on $T$, for when these characterizations are first-order expressible, i.e., for when a model companion of each $T^C_θ$ exists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_13667 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Model Theory of Generic Vector Space Endomorphisms Chini, Leon Logic 03C10 (Primary) 03C60 (Secondary) This paper deals with 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 then consider 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 $\operatorname{Ker}(θ^n) = \operatorname{Ker}(θ^{n+1})$ can be expressed in such a manner. We then parametrize the consistent extensions of this form by a family $\{T^C_θ: C \in \mathcal{C}\}$ and characterize the existentially closed models of each $T^C_θ$. We also present a sufficient criterion, which only depends on $T$, for when these characterizations are first-order expressible, i.e., for when a model companion of each $T^C_θ$ exists. |
| title | Model Theory of Generic Vector Space Endomorphisms |
| topic | Logic 03C10 (Primary) 03C60 (Secondary) |
| url | https://arxiv.org/abs/2502.13667 |