Towards a Finer Classification of Strongly Minimal Sets
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2021
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909184314834944 |
|---|---|
| author | Baldwin, John T. Verbovskiy, Viktor V. |
| author_facet | Baldwin, John T. Verbovskiy, Viktor V. |
| contents | Let $M$ be strongly minimal and constructed by a `Hrushovski construction'. If the Hrushovski algebraization function $μ$ is in a certain class ${\mathcal T}$ ($μ$ triples) we show that for independent $I$ with $|I| >1$, ${\rm dcl}^*(I)= \emptyset$ (* means not in ${\rm dcl}$ of a proper subset). This implies the only definable truly $n$-ary function $f$ ($f$ `depends' on each argument), occur when $n=1$.
We prove, indicating the dependence on $μ$, for Hrushovski's original construction and including analogous results for the strongly minimal $k$-Steiner systems of Baldwin and Paolini 2021 that the symmetric definable closure, ${\rm sdcl}^*(I) =\emptyset$, and thus the theory does not admit elimination of imaginaries. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if $k = p^n$. The proofs depend on our introduction for appropriate $G \subseteq {\rm aut}(M)$ the notion of a $G$-normal substructure ${\mathcal A}$ of $M$ and of a $G$-decomposition of ${\mathcal A}$.
These results lead to a finer classification of strongly minimal structures with flat geometry; according to what sorts of definable functions they admit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_15567 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Towards a Finer Classification of Strongly Minimal Sets Baldwin, John T. Verbovskiy, Viktor V. Logic 03C05 (Primary) 03C45, 03C98 (Secondary) Let $M$ be strongly minimal and constructed by a `Hrushovski construction'. If the Hrushovski algebraization function $μ$ is in a certain class ${\mathcal T}$ ($μ$ triples) we show that for independent $I$ with $|I| >1$, ${\rm dcl}^*(I)= \emptyset$ (* means not in ${\rm dcl}$ of a proper subset). This implies the only definable truly $n$-ary function $f$ ($f$ `depends' on each argument), occur when $n=1$. We prove, indicating the dependence on $μ$, for Hrushovski's original construction and including analogous results for the strongly minimal $k$-Steiner systems of Baldwin and Paolini 2021 that the symmetric definable closure, ${\rm sdcl}^*(I) =\emptyset$, and thus the theory does not admit elimination of imaginaries. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if $k = p^n$. The proofs depend on our introduction for appropriate $G \subseteq {\rm aut}(M)$ the notion of a $G$-normal substructure ${\mathcal A}$ of $M$ and of a $G$-decomposition of ${\mathcal A}$. These results lead to a finer classification of strongly minimal structures with flat geometry; according to what sorts of definable functions they admit. |
| title | Towards a Finer Classification of Strongly Minimal Sets |
| topic | Logic 03C05 (Primary) 03C45, 03C98 (Secondary) |
| url | https://arxiv.org/abs/2106.15567 |