Dense codense predicates and the NTP<sub>2</sub>

Based on the work done in in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking‐independent elements that is dense inside a partial type G(x), which we call H‐structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again H‐structures. We prove that under these assumptions the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one‐basedness and CM‐triviality. In the one‐based case, when T has SU‐rank ωα and the SU‐rank is continuous, we take G(x) to be the type of elements of SU‐rank ωα and we describe a natural “geometry of generics modulo H” associated with such expansions and show it is modular.

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“…The following definition and proposition were very fruitfull to show the preservation of NTP2 to T ind when T was geometric; cf. . Definition Let

false(M, H false) ⊧ T_{scriptG}^{ind}

be saturated.…”

Section: Definable Sets In H‐structuresmentioning

confidence: 99%

“…We also get by transitivity that a n+1 | Ca 1 ,...,a n H 1 The following definition and proposition were very fruitfull to show the preservation of NTP2 to T ind when T was geometric; cf. [3].…”

Section: P R O O Fmentioning

confidence: 99%

Supersimple structures with a dense independent subset

Berenstein

Carmona

Vassiliev

2017

Mathematical Logic Qtrly

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“…The approach we follow is to check the property by doing a formula-by-formula analysis, separating the cases when the corresponding definable set is small (algebraic over the predicate) or large. To show the preservation of NSOP 1 we build on ideas presented in [23], the proofs for the preservation of NTP 1 , NTP 2 generalize ideas presented in [2,18]. In Section 6, we study independence notions in the expansion, assuming the original theory has a good notion of independence.…”

Section: Introductionmentioning

confidence: 99%

Vector spaces with a dense-codense generic submodule

Berenstein¹,

d’Elbée²,

Vassiliev³

2022

Preprint

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We study expansions of a vector space V over a field F, possibly with extra structure, with a generic submodule over a subring of F. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP 1 , NTP 2 and NSOP 1 . We also study induced independence relations in the expansion.

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“…The study of expansions in the NTP 2 context was undertaken in , where it was shown that the NTP 2 property is preserved under “dense and codense” unary predicate expansions of geometric structures where the unary predicate is assumed to define either an algebraically independent subset or an elementary substructure. In the present paper, we prove that the NTP 1 property is also preserved under such expansions.…”

Section: Introductionmentioning

confidence: 99%

“…Our paper is organized as follows. In § , we review some essential facts about dense codense predicate expansions from . In § , we state slightly modified versions of some results from concerning the SOP 2 , the strong order property of order 2 (equivalently, the TP 1 ).…”

Section: Introductionmentioning

confidence: 99%

A preservation theorem for theories without the tree property of the first kind

Dobrowolski

Kim

2017

Mathematical Logic Qtrly

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We prove that the NTP 1 property of a geometric theory T is inherited by theories of lovely pairs and H-structures associated to T . We also provide a class of examples of nonsimple geometric NTP 1 theories. IntroductionOne theme of research in model theory is to inquire whether some well-known properties are preserved under a certain unary predicate expansions of a given structure. One of the motivations for this is that positive theorems of this kind often allows us to obtain interesting and complicatedlooking theories which still satisfy some strong tameness conditions. The study of expansions by unary predicates reaches back to the paper of Poizat on beautiful pairs [10], and has been developed in various directions ever since. Remarkable papers on this subject include for example [5], where the stability condition is examined, and [1], which is dedicated to studying expansions in simple theories (which generalize the stable ones).The well-known equivalence TP ⇔ TP 1 ∨ TP 2 due to Shelah [11] (where TP denotes the tree property while TP 1 and TP 2 denote the tree properties of the first and second kind, respectively), suggests two natural generalizations of simple theories, namely NTP 1 theories and NTP 2 theories (i.e., theories without TP 1 and TP 2 , respectively). So far, NTP 1 and NTP 2 theories have been studied much less extensively than the simple ones (i.e. theories without TP). However, recently some interesting results on these theories began to appear, notably [6] and [7]. In particular, natural examples of non-simple NTP 1 thoeries were provided in [6], namely: ω-free PAC fields, linear spaces with a generic bilinear form and a class of theories obtained by the "pfc" construction.The study of expansions in the NTP 2 context was undertaken in [2], where it was shown that the NTP 2 property is preserved under "dense and codense" unary predicate expansions of geometric structures where the unary predicate is assumed to define either an algebraically independent subset or an elementary substructure. In the present paper, we prove that the NTP 1 property is also preserved under such expansions. One of the main ingredients in our proof is the recently proved fact (due to Chernikov and Ramsey [6]) that the TP 1 property can, in any TP 1 theory, always be witnessed by some formula in a single free variable. We also prove (in Section 4) that an NTP 1 nonsimple geometric theory can be obtained from any Fraïssé limit which has a simple *

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Dense codense predicates and the NTP₂

Abstract: Abstract. We show that if T is any geometric theory having NTP 2 then the corresponding theories of lovely pairs of models of T and of H-structures associated to T also have NTP 2 . We also prove that if T is strong then the same two expansions of T are also strong.

Cited by 6 publications

References 20 publications

Supersimple structures with a dense independent subset

Supersimple structures with a dense independent subset

Vector spaces with a dense-codense generic submodule

A preservation theorem for theories without the tree property of the first kind

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Dense codense predicates and the NTP2

Abstract: Abstract. We show that if T is any geometric theory having NTP 2 then the corresponding theories of lovely pairs of models of T and of H-structures associated to T also have NTP 2 . We also prove that if T is strong then the same two expansions of T are also strong.

Cited by 6 publications

References 20 publications

Supersimple structures with a dense independent subset

Supersimple structures with a dense independent subset

Vector spaces with a dense-codense generic submodule

A preservation theorem for theories without the tree property of the first kind

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Product

Resources

About

Dense codense predicates and the NTP₂