Quasiminimal structures and excellence

Bays, Martin; Hart, Bradd; Hyttinen, Tapani; Kesälä, Meeri; Kirby, Jonathan

doi:10.1112/blms/bdt076

Cited by 25 publications

(65 citation statements)

References 11 publications

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“…Set (i) C 0 (H) ⊆ C 0 (M) as classes with embeddings and (ii) for every finite X ⊆ H there is N ∈ C 0 (H), such that X ⊆ N. Proof. This follows from 2.17 by the main result of [16].…”

Section: 17mentioning

confidence: 59%

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Analytic Zariski structures and non-elementary categoricity

Zilber¹

2017

Beyond First Order Model Theory

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We study analytic Zariski structures from the point of view of non-elementary model theory. We show how to associate an abstract elementary class with a one-dimensional analytic Zariski structure and prove that the class is stable, quasi-minimal and homogeneous over models. We also demonstrate how Hrushovski's predimension arises in this general context as a natural geometric notion and use it as one of our main tools.

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Section: 17mentioning

confidence: 59%

“…], for this construction. Similar construction was used in [16]. Set (i) C 0 (H) ⊆ C 0 (M) as classes with embeddings and (ii) for every finite X ⊆ H there is N ∈ C 0 (H), such that X ⊆ N. Proof.…”

Section: 17mentioning

confidence: 99%

Analytic Zariski structures and non-elementary categoricity

Zilber¹

2017

Beyond First Order Model Theory

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“…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. There, the canonical pregeometry is obtained from the bounded closure operator, which corresponds to the algebraic closure operator in the first order case.…”

mentioning

confidence: 99%

“…We will work in the context of quasiminimal classes, i.e. 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%

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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“…We do not elaborate on the definition any further. The interested reader can consult [2,7,19]. Note however that given a strongly regular type p in an arbitrary theory, the class of elementary submodels of the monster model C (closed under isomorphisms) together with the restriction of cl p satisfies the first two axioms (Closure under isomorphisms and Quantifier free theory).…”

Section: ω-Stable Theoriesmentioning

confidence: 99%

Constructing quasiminimal structures

Haykazyan

2017

Mathematical Logic Qtrly

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Quasiminimal structures play an important role in non-elementary categoricity. In this paper we explore possibilities of constructing quasiminimal models of a given first-order theory. We present several constructions with increasing control of the properties of the outcome using increasingly stronger assumptions on the theory. We also establish an upper bound on the Hanf number of the existence of arbitrarily large quasiminimal models.

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Quasiminimal structures and excellence

Abstract: We show that the excellence axiom in the definition of Zilber's quasiminimal excellent classes is redundant, in that it follows from the other axioms. This substantially simplifies a number of categoricity proofs.

Cited by 25 publications

References 11 publications

Analytic Zariski structures and non-elementary categoricity

Analytic Zariski structures and non-elementary categoricity

Finding a field in a Zariski-like structure

Constructing quasiminimal structures

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