On algebraically closed fields with a distinguished subfield

d’Elbée, Christian; Kaplan, Itay

doi:10.48550/arxiv.2108.04160

Cited by 1 publication

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“…In [DKL22], d'Elbée, Kaplan, and Neuhauser gave preservation results on a pair of an algebraically closed field and its distinguished subfield. Namely, if the distinguished subfield is stable, NIP, or NSOP 1 , then so is the pair structure.…”

Section: The Pair Of An Algebraically Closed Field and Its Distinguis...mentioning

confidence: 99%

“…In this section, we aim to prove that if the distinguished subfield is NTP 1 , NTP 2 , or NATP, then so is the pair structure. Our approach is largely motivated by the proof of the preservation of NIP in [DKL22,Theorem 5.33]. We first recall several basic facts and notions on a pair of an algebraically closed field and its distinguished subfield from [DKL22].…”

Section: The Pair Of An Algebraically Closed Field and Its Distinguis...mentioning

confidence: 99%

“…Our approach is largely motivated by the proof of the preservation of NIP in [DKL22,Theorem 5.33]. We first recall several basic facts and notions on a pair of an algebraically closed field and its distinguished subfield from [DKL22].…”

Section: The Pair Of An Algebraically Closed Field and Its Distinguis...mentioning

confidence: 99%

“…The paper [DKL22] gave preservation results on a pair of an algebraically closed field and its distinguished subfield; there, it is shown that the pair structure is stable, NIP, or NSOP 1 if the distinguished subfield has the corresponding one. In Section 6, motivated by [DKL22], we further investigate preservation of model theoretic tree properties: If the distinguished subfield is NTP 1 , NTP 2 , or NATP, then so is the pair structure. As a colrollary, we conclude hat a PAC field is NATP (and NTP 1 respectively) if and only if the Galois group of the field has the complete system of NATP (and NTP 1 respectively).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Pair Of An Algebraically Closed Field and Its Distinguis...mentioning

confidence: 99%

Section: The Pair Of An Algebraically Closed Field and Its Distinguis...mentioning

confidence: 99%