Quasiminimal structures, groups and Zariski-like geometries

Hyttinen, Tapani; Kangas, Kaisa

doi:10.1016/j.apal.2016.02.002

Cited by 9 publications

(72 citation statements)

References 24 publications

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Mathematics subject classification: 03C50, 03C98Definition 2.7. We say that a tuple a is Galois definable from a set A, if it holds for every f ∈ Aut(M/A) that f (a) = a.

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confidence: 99%

“…In [6], we presented Zariski-like quasiminimal pregeometry structures as nonelementary generalizations of Zariski geometries. Quasiminimal pregeometry structures (in sense of [1]) provide a non-elementary analogue for strongly minimal structures from the first order context.…”

mentioning

confidence: 99%

“…abstract elementary classes (AECs) that arise from a quasiminimal pregeometry structure (see [1]). These classes are uncountably categorical and have both AP and JEP and thus also a universal model homogeneous monster model which we will denote M. By [6], they also have a perfect theory of independence.…”

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confidence: 99%

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Finding a field in a Zariski-like structure

Kangas

2017

Annals of Pure and Applied Logic

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We show that if M is a Zariski-like structure (see [6]) and the canonical pregeometry obtained from the bounded closure operator (bcl) is non locally modular, then M interprets either an algebraically closed field or a non-classical group. Mathematics subject classification: 03C50, 03C98Definition 2.7. We say that a tuple a is Galois definable from a set A, if it holds for every f ∈ Aut(M/A) that f (a) = a. We write a ∈ dcl(A), and say that a is in the definable closure of A.We say that a and b are interdefinable if a ∈ dcl(b) and b ∈ dcl(a). We say that they are interdefinable over A if a ∈ dcl(Ab) and b ∈ dcl(Aa).It turns out that in our setting, bounded Galois definable sets are countable (by Lemmas 2.24 and 2.26 in [6]).Our main notion of type will be that of the weak type rather than the Galois type:Definition 2.9. Let A ⊂ M. We say b and c have the same weak type over A,An analogue for the strong types of the first-order context is provided by Lascar types.Definition 2.10. Let A be a finite set, and let E be an equivalence relation on M n , for some n < ω. We sayWe denote the set of all A-invariant equivalence relations that have only boundedly many equivalence classes by E(A).We say that a and b have the same Lascar type over a set B, denoted Lt(a/B) = Lt(b/B), if for all finite A ⊆ B and all E ∈ E(A), it holds that (a, b) ∈ E. Remark 2.11. By Lemma 2.37 in [6], Lascar types imply weak types. Moreover, Lascar types are stationary by Lemma 2.46 in [6].If p is a stationary type over A and A ⊂ C, we write p| C for the (unique) free extension of p into C.Definition 2.12. Let B ⊂ M. We say an element b ∈ B is generic over some set A if dim(b/A) is maximal (among the elements of B). The set A is not mentioned if it is clear from the context. For instance, if B is assumed to be Galois definable over some set D, then we usually assume A = D.Let p = t(a/A) for some a ∈ M and A ⊂ B. We say b ∈ M is a generic realization of p (over B) if dim(b/B) is maximal among the realizations of p.Definition 2.13. We say that a sequence (a i ) i<α is indiscernible over A if every permutation of the sequence {a i | i < α} extends to an automorphism f ∈ Aut(M/A).We say a sequence (a i ) i<α is strongly indiscernible over A if for all cardinals κ, there are a i , α ≤ i < κ, such that (a i ) i<κ is indiscernible over A.

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“…

Mathematics subject classification: 03C50, 03C98Definition 2.7. We say that a tuple a is Galois definable from a set A, if it holds for every f ∈ Aut(M/A) that f (a) = a.

…”

mentioning

confidence: 99%

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confidence: 99%

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Finding a field in a Zariski-like structure

Kangas

2017

Annals of Pure and Applied Logic

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“…in [Kir10,§4], see also [HK16,2.87]) to see that any quasiminimal pregeometry class is a quasiminimal AEC, but here we prove a converse (Theorem 4.17). We have to solve two difficulties:…”

Section: Introductionmentioning

confidence: 85%

Quasiminimal abstract elementary classes

Vasey

2017

Arch. Math. Logic

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Abstract. We propose the notion of a quasiminimal abstract elementary class (AEC). This is an AEC satisfying four semantic conditions: countable Löwenheim-Skolem-Tarski number, existence of a prime model, closure under intersections, and uniqueness of the generic orbital type over every countable model. We exhibit a correspondence between Zilber's quasiminimal pregeometry classes and quasiminimal AECs: any quasiminimal pregeometry class induces a quasiminimal AEC (this was known), and for any quasiminimal AEC there is a natural functorial expansion that induces a quasiminimal pregeometry class. We show in particular that the exchange axiom is redundant in Zilber's definition of a quasiminimal pregeometry class.

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“…, which also contains the result that we shall apply to show that the models in

K^{*}

are either trivial or essentially vector spaces. Other examples of these generalisations include, e.g., and together with . The paper contains concrete examples of non‐elementary classes in which these results can be applied.…”

Section: Introductionmentioning

confidence: 99%

Categoricity and universal classes

Hyttinen

Kangas

2018

Mathematical Logic Qtrly

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Let (K,⊆) be a universal class with LS (K)=λ categorical in a regular κ>λ+ with arbitrarily large models, and let K∗ be the class of all A∈scriptK>λ for which there is B∈scriptK≥κ such that A⊆B. We prove that K∗ is totally categorical (i.e., ξ‐categorical for all ξ> LS (K)) and scriptK≥ℶμ+⊆scriptK∗ for μ=2λ+. This result is partially stronger and partially weaker than a related result due to Vasey. In addition to small differences in our categoricity transfer results, we provide a shorter and simpler proof. In the end we prove the main theorem of this paper: the models of K>λ+∗ are essentially vector spaces (or trivial, i.e., disintegrated).

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Quasiminimal structures, groups and Zariski-like geometries

Cited by 9 publications

References 24 publications

Finding a field in a Zariski-like structure

Finding a field in a Zariski-like structure

Quasiminimal abstract elementary classes

Categoricity and universal classes

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