Forking and Dividing in NTP<sub>2</sub> theories

Chernikov, Artem; Kaplan, Itay

doi:10.2178/jsl/1327068688

Cited by 63 publications

(124 citation statements)

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“…Now g · X is also closed in G. So by results in [5] and [13] (see also [22]), g · X forks over M 0 . By the main result of [2] (which is maybe implicit in other papers in the o-minimal case), g · X divides over M 0 . As X is defined over M 0 this means that for some M 0 -indiscernible sequence (g i : i < ω) and some k < ω, {g i · X : i < ω} is k-inconsistent, in the sense that for every (some) i 1 < .. < i k , (g i 1 · X) ∩ .... ∩ (g i k · X) = ∅.…”

Section: Definition 41 Supposeκ Is An Inaccessible Cardinal Andm Amentioning

confidence: 83%

Connected components of definable groups, and o-minimality II

Conversano

Pillay

2015

Annals of Pure and Applied Logic

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We give examples of definable groups G (in a saturated model, sometimes o-minimal) such that G 00 = G 000 , yielding also new examples of "non G-compact" theories. We also prove that for G definable in a (saturated) o-minimal structure, G has a "bounded orbit" (i.e. there is a type of G whose stabilizer has bounded index) if and only if G is definably amenable, giving a positive answer to a conjecture of Newelski and Petrykowski in this special case of groups definable in ominimal structures. We also introduce and discuss further conjectures on bounded orbits in definable groups. These results and analyses are informed by a decomposition theorem for groups in o-minimal structures.

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Section: Definition 41 Supposeκ Is An Inaccessible Cardinal Andm Amentioning

confidence: 83%

Connected components of definable groups, and o-minimality II

Conversano

Pillay

2015

Annals of Pure and Applied Logic

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“…Combined with the results on forking and dividing in NIP theories from [7], we deduce the following: working over a model M , let {φ(x, a) : a |= q(y)} be a family of non-forking instances of φ(x, y), where the parameter a ranges over the set of solutions of a partial type q. Combined with the results on forking and dividing in NIP theories from [7], we deduce the following: working over a model M , let {φ(x, a) : a |= q(y)} be a family of non-forking instances of φ(x, y), where the parameter a ranges over the set of solutions of a partial type q.…”

Section: Introductionmentioning

confidence: 91%

Externally definable sets and dependent pairs II

Chernikov

Simon

2015

Trans. Amer. Math. Soc.

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We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; over a model, a family of non-forking instances of a formula (with parameters ranging over a type-definable set) can be covered with finitely many invariant types; we give some criteria for the boundedness of an expansion by a new predicate in a distal theory; naming an arbitrary small indiscernible sequence preserves NIP, while naming a large one doesn't; there are models of NIP theories over which all 1-types are definable, but not all n-types.

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“…We will need some facts about forking and dividing in NTP 2 theories established in [CK12]. Recall that a set C is an extension base if every type in S(C) does not fork over C.…”

Section: Ntp 2 Implies the Chain Conditionmentioning

confidence: 99%

“…The realization that it is possible to develop a good theory of forking in the NTP 2 context came from the paper [CK12], where it was demonstrated that the basic theory can be carried out as long as one is working over an extension base (a set is called an extension base if every complete type over it has a global non-forking extension, e.g. any model or any set in a simple, o-minimal or C-minimal theory is an extension base).…”

Section: Introductionmentioning

confidence: 99%

An Independence Theorem for Ntp2 Theories

Yaacov

Chernikov²

2014

J. symb. log.

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Abstract. We establish several results regarding dividing and forking in NTP 2 theories.We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski's terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP 2 theory.We also define the dividing order of a theory -a generalization of Poizat's fundamental order from stable theories -and give some equivalent characterizations under the assumption of NTP 2 . The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy.

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Forking and Dividing in NTP₂ theories

Cited by 63 publications

References 8 publications

Connected components of definable groups, and o-minimality II

Connected components of definable groups, and o-minimality II

Externally definable sets and dependent pairs II

An Independence Theorem for Ntp2 Theories

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Forking and Dividing in NTP2 theories

Cited by 63 publications

References 8 publications

Connected components of definable groups, and o-minimality II

Connected components of definable groups, and o-minimality II

Externally definable sets and dependent pairs II

An Independence Theorem for Ntp2 Theories

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Forking and Dividing in NTP₂ theories