Distality in valued fields and related structures

Aschenbrenner, Matthias; Chernikov, Artem; Gehret, Allen; Ziegler, Martin

doi:10.1090/tran/8661

Cited by 18 publications

(24 citation statements)

References 52 publications

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“…In particular, my gratitude to both of them for sharing with me their ideas which led to the proof of Proposition 2.4 and motivated the writing of this paper. Many thanks to Artem Chernikov and Martin Ziegler for sharing their results in [ACGZ18]. Finally, many thanks to Martin Bays and Allen Gehret, whose help and discussions have greatly improved the quality of this paper.…”

Section: Acknowledgementsmentioning

confidence: 92%

Stably embedded submodels of Henselian valued fields

Touchard

2020

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We reduce the question of definability of types over a submodel of certain Henselian valued fields to the corresponding question in the value group and residue field. Then, we treat the question of the axiomatization of stably embedded elementary pairs of these Henselian valued fields. This work extends previous results of Cubides and Delon on algebraically closed valued fields and results of Cubides and Ye on real closed valued fields.

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Section: Acknowledgementsmentioning

confidence: 92%

Stably embedded submodels of Henselian valued fields

Touchard

2020

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“…The model theory of ordered abelian groups has been studied since the sixties, and several results regarding quantifier elimination and the model complexity of certain classes of ordered abelian groups have been obtained. See for example [5] [11] [12] and [9]. The next natural step regarding the model theory of ordered abelian groups was understanding a reasonable language where one will have elimination of imaginaries.…”

Section: Introductionmentioning

confidence: 99%

Elimination of imaginaries in Ordered Abelian groups with bounded regular rank

Mariana¹

2021

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In this paper we study elimination of imaginaries in some classes of pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups Γ ∆ for each definable convex subgroup ∆, and sorts for the quotient groups Γ ∆ + lΓ where ∆ is a definable convex subgroup and l ∈ N≥2. We refer to these sorts as the quotient sorts. For the dp-minimal case we obtain a complete elimination of imaginaries, if we also add constants to distinguish the cosets of ∆ + nΓ in Γ, where ∆ is a definable convex subgroup and n ∈ N≥2.Aknowledgements: This document was written as part of the author's PhD thesis under Pierre Simon and Thomas Scanlon. The author would like to express her gratitude to both of them, for many insightful conversations and their time. The author would like to thank particularly Pierre for his continuous encouragement and his financial support from his NSF Grant 1848562. The author would like to thank as well R. Mennuni for some useful comments in a previous draft. Preliminaries Elimination of imaginariesLet T be a first order theory and M be its monster model. Let D ⊆ M k be some definable set and E some definable equivalence relation over D. The equivalence class e = a E is said to be an imaginary element. Imaginaries in model theory were introduced by Shelah in [14]. Later in [15], Makkai proposed to construct the many sorted structure M eq , where we add a sort S E for each definable equivalence relation E and a map π E sending each element to its class. Since then, the model theoretic community has presented and studied imaginary elements in this way and refer to the multi-sorted structure M eq , as the imaginary expansion of M. We call the sorts S E as imaginary sorts while we refer to M as the home-sort. Any formula φ(x, ȳ) induces an equivalence relation in M ȳ defined asLet b ∈ M ȳ and X ∶= φ(x, b). We call the class b E φ the code of X and denote it as ⌜X⌝. We denote as dcl eq and acl eq the definable closure and the algebraic closure in the expansion M eq . Definition 2.1.1. We say that T has elimination of imaginaries if for any imaginary element e there is a tuple a in the home-sort such that e ∈ dcl eq (a) and a ∈ dcl eq (e).2. We say that T has weak elimination of imaginaries if for any imaginary element e there is a tuple a in the home-sort such that e ∈ dcl eq (a) and a ∈ acl eq (e).3. We say that T codes finite sets if for every model M ⊧ T , and every set S of a finitely many elements in M , the code ⌜S⌝ is interdefinable with a tuple of elements in M .The following is a folklore fact.Fact 2.2. Let T be a complete multi-sorted theory. If T has weak elimination of imaginaries and codes finite sets then T eliminates imaginaries.

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“…However, in general no angular map is definable in the traditional language of valued fields; and naming such a map changes the structure 1 . Only recently [2,Theorem 5.15] a language was found where the theory of Henselian valued fields of equicharacteristic 0 (with no named angular component) admits quantifier elimination relative to the residue field and value group. The authors of [2] use first a result of Flenner [6] -which eliminates 2020Mathematics Subject Classification: Primary: 03C10 03C45; Secondary: 03C60, 15A03 * The author was partially supported through the "Oberwolfach Leibniz Fellows" program -Project-ID2043r, by Mathematisches Forschungsinstitut Oberwolfach in 2020 and supported by a grant of the University of Campania 'Luigi Vanvitelli' in the framework of V:ALERE 2019 (GoAL project) 1 for instance, the ultraproduct U Qp has burden 1 in the language of valued fields, and 2 in the language of Pas.…”

Section: Introductionmentioning

confidence: 99%

“…quantifiers relative to the RV-sort 2 . Then with a clever use of p.p.-elimination of quantifiers in abelian structures, they describe a natural language where this RV-sort eliminates quantifiers relative to the value group and the residue field.…”

Section: Introductionmentioning

confidence: 99%

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On Model Theory of Valued Vector Spaces

Touchard¹

2021

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In analogy to valued fields, we study model-theoretic properties of valued vector spaces with variable base field by proving transfer principles down to the skeleton and down to the value set and base field. For instance, we give a formula which computes its burden in terms of the burden of its base field and its value set. To do this, we study these transfer principles in the context of lexicographic products of structures.

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Distality in valued fields and related structures

Cited by 18 publications

References 52 publications

Stably embedded submodels of Henselian valued fields

Stably embedded submodels of Henselian valued fields

Elimination of imaginaries in Ordered Abelian groups with bounded regular rank

On Model Theory of Valued Vector Spaces

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