Generic variations and NTP $$_1$$ 1

Dobrowolski, Jan

doi:10.1007/s00153-018-0609-4

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…Finally, the fact that T * feq2 is NSOP 1 follows from an unpublished result of Ramsey that NSOP 1 is preserved by the generic variation construction from [3] (the analogous statement for NTP 1 was shown by Dobrowolski in [10]).…”

Section: Examplesmentioning

confidence: 70%

Remarks on generic stability in independent theories

Conant

Gannon

2020

Annals of Pure and Applied Logic

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In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of "generic stability" in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as φ-types in simple theories that are definable and finitely satisfiable in a small model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs.Date: May 28, 2019.

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Section: Examplesmentioning

confidence: 70%

Remarks on generic stability in independent theories

Conant

Gannon

2020

Annals of Pure and Applied Logic

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“…In [CP98], the problem of adding a generic unary predicate symbol was studied, where such an expansion is called the generic predicate construction. We know that NTP 2 is preserved by the generic predicate construction by [Che14] and NTP 1 by [Dob18], whose idea of the proof is similar as [Che14]. Utilizing ideas there, we show in Section 5 that NATP is preserved under such a construction.…”

Section: Introductionmentioning

confidence: 81%

Preservation of NATP

Ahn¹,

Kim²,

Lee³

et al. 2022

Preprint

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We prove several preservation theorems for NATP and furnish several examples of NATP.First, we prove preservation of NATP for the parametrization and sum of the theories of Fraïssé limits of Fraïssé classes satisfying strong amalgamation property.Second, we prove preservation of NATP for two kinds of dense/co-dense expansions, that is, the theories of lovely pairs and of H-structures for geometric theories and dense/co-dense expansion on vector spaces.Third, we prove preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also preservations of NTP 1 and NTP 2 are considered.Fourth, we present some proper examples of NATP using the results proved in this paper. Most of all, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP.

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“…Finally, T ∩ is interpretable in the theory of equality. Dobrowolski [Dob18] showed that T * Var is NTP 1 when T is NTP 1 . It is conjectured that NSOP 1 and NTP 1 are equivalent.…”

Section: Examplesmentioning

confidence: 99%

Interpolative fusions II: Preservation results

Kruckman¹,

Tran²,

Walsberg³

2022

Preprint

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We study interpolative fusion, a method of combining theories T 1 and T 2 in distinct languages in a "generic" way over a common reduct T∩, to obtain a theory T * ∪ . When each T i is model-complete, T * ∪ is the model companion of the union T 1 ∪ T 2 . Our goal is to prove preservation results, i.e., to find sufficient conditions under which model-theoretic properties of T 1 and T 2 are inherited by T * ∪ . We first prove preservation results for quantifier elimination, modelcompleteness, and related properties. We then apply these tools to show that, under mild hypotheses, including stability of T∩, the property NSOP 1 is preserved. We also show that simplicity is preserved under stronger hypotheses on algebraic closure in T 1 and T 2 . This generalizes many previous results; for example, simplicity of ACFA and the random n-hypergraph are both non-obvious corollaries. We also address preservation of stability, NIP, and ℵ 0 -categoricity, and we describe examples which witness that these results are sharp.

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Generic variations and NTP $$_1$$ 1

Abstract: We prove a preservation theorem for NTP 1 in the context of the generic variations construction. We also prove that NTP 1 is preserved under adding to a geometric theory a generic predicate.

Cited by 3 publications

References 9 publications

Remarks on generic stability in independent theories

Remarks on generic stability in independent theories

Preservation of NATP

Interpolative fusions II: Preservation results

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