Independence over arbitrary sets in NSOP1 theories

Dobrowolski, Jan; Kim, Byunghan; Ramsey, Nicholas

doi:10.1016/j.apal.2021.103058

Cited by 7 publications

(43 citation statements)

References 16 publications

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“…We say that a theory T satisfies the existence axiom (or simply existence) if every set of parameters is an extension base. It was asked in [5, Question 6.6] whether any NSOP 1 theory satisfies existence, and a list of positive examples was given in [5,Fact 2.14]. Here we show that T ∞ also satisfies it:…”

Section: Independence Relations and Genericsmentioning

confidence: 69%

“…Kim-independence was introduced and studied extensively in [14] over models in NSOP 1 theories. It was proved there, among other results, that | K is symmetric and satisfies the independence theorem over models, which was later extended in [5] to arbitrary sets in NSOP 1 theories satisfying existence. (1) We say a formula ϕ(x, a 0 ) Kim-divides over A if for some Morley sequence Example 8.11.…”

Section: Independence Relations and Genericsmentioning

confidence: 92%

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Sets, groups, and fields definable in vector spaces with a bilinear form

Dobrowolski¹

2023

Annales De l'Institut Fourier

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We study definable sets, groups, and fields in the theory of infinite-dimensional vector spaces over an algebraically closed field of any fixed characteristic different from 2 equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define a notion of dimension of a definable set, which enjoys many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all definable groups are (algebraic-by-abelian)-by-algebraic. We conclude that every definable field is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component.We also prove analogous results working over real closed fields.Résumé. -Nous étudions des ensembles, des groupes et des corps définissables dans la théorie des espaces vectoriels de dimension infinie sur un corps algébriquement clos de caractéristique différente de 2 munis d'une forme bilinéaire symétrique (ou alternée) non dégénérée. Tout d'abord, nous définissons une notion de dimension d'un ensemble définissable, qui possède de nombreuses propriétés de rang de Morley dans les théories fortement minimales. Ensuite, en utilisant cette notion de dimension comme outil principal, nous prouvons que tous les groupes définissables sont (algébriques-par-abéliens)-par-algébriques. Nous concluons que tout corps définissable est définissablement isomorphe au corps des scalaires de l'espace vectoriel. Nous déduisons d'autres conséquences du bon comportement de la dimension, par exemple chaque type générique dans tout ensemble définissable est un type définissable ; chaque ensemble est une base d'extension ; chaque groupe définissable a une composante connexe définissable.Nous démontrons également des résultats analogues en travaillant sur des corps réels clos.

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Section: Independence Relations and Genericsmentioning

confidence: 69%

Section: Independence Relations and Genericsmentioning

confidence: 92%

Sets, groups, and fields definable in vector spaces with a bilinear form

Dobrowolski¹

2023

Annales De l'Institut Fourier

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“…There it is shown that, over a model, that notion is equivalent to the one stated in Definition 3.1 (3). Since in general even in a simple theory, there need not exist a global invariant extension of a type over a set, instead in [6] the above definition in (3) is coherently given as Kim-dividing over an arbitrary set.…”

Section: Kim-forking and Tpmentioning

confidence: 99%

“…As we will not deal with these facts, see [2] or [11] for more details. Further advances are discovered in [7], [8], [6], [4] recently. Namely, it is shown that in any NSOP 1 T having nonforking existence (as said any simple T , and all the known NSOP 1 T have this), the notions of Kim-forking and Kim-dividing coincide, and ⌣ | K supplies a good independence notion since it satisfies all the aforementioned properties that hold of ⌣ | in simple theories, except base monotonicity (so there can exist d and…”

Section: And Nonforking Existence (That Is: D ⌣ |mentioning

confidence: 99%

“…Recently in [7], [8], it is shown that in any NSOP 1 theory, over models, 'Kim-forking' satisfies all the aforementioned axioms that forking satisfies in simple theories, except base monotonicity (one direction of full transitivity). Then it is proved in [6], [4] that the same axioms hold over arbitrary sets in any NSOP 1 theory having nonforking existence. So far summarized results justify our study of various tree properties in this paper.…”

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confidence: 99%

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More on tree properties

Casanovas

Kim

2020

Fund. Math.

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Tree properties are introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP 1 or TP 2 . In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion as it satisfies symmetry, full transitivity, extension, local character, and typeamalgamation, over sets. Shelah also introduced SOP n (n-strong order property). Recently it is proved that in any NSOP 1 theory (i.e. a theory not having SOP 1 ) holding nonforking existence, Kim-forking also satisfies all the mentioned independence properties except base monotonicity (one direction of full transitivity). These results are the sources of motivation for this paper.Mainly, we produce type-counting criteria for SOP 2 (which is equivalent to TP 1 ) and SOP 1 . In addition, we study relationships between TP 2 and Kim-forking, and obtain that a theory is supersimple iff there is no countably infinite Kim-forking chain.

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Generic Stability Independence and Treeless Theories

Kaplan,

Ramsey,

Simon

2024

Forum of Mathematics, Sigma

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We initiate a systematic study of generic stability independence and introduce the class of treeless theories in which this notion of independence is particularly well behaved. We show that the class of treeless theories contains both binary theories and stable theories and give several applications of the theory of independence for treeless theories. As a corollary, we show that every binary NSOP $_{3}$ theory is simple.

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Independence over arbitrary sets in NSOP1 theories

Cited by 7 publications

References 16 publications

Sets, groups, and fields definable in vector spaces with a bilinear form

Sets, groups, and fields definable in vector spaces with a bilinear form

More on tree properties

Generic Stability Independence and Treeless Theories

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