On <i>n</i>‐dependent groups and fields

Hempel, Nadja

doi:10.1002/malq.201400080

Cited by 20 publications

(36 citation statements)

References 16 publications

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“…Quasifinite theories are studied in depth in [CH03], and in [Hru13, Section 6.5] it is pointed out that every quasi-finite theory is 2-dependent: it is demonstrated in [CH03] using the classification of finite simple groups that in a quasifinite theory, π ∆ (m) grows at most as 2 m (see Definition 3.10 and Proposition 6.5). An example of a quasifinite theory is the theory of a generic bilinear form on an infinite-dimensional vector space over a finite field (a direct proof that this theory is 2-dependent is given in [Hem14]).…”

Section: A Theorymentioning

confidence: 99%

“…In [She07] Shelah demonstrates some results about connected components for (type)-definable groups in 2-dependent theories (which can be viewed as a form of modularity in certain context, see remarks in [Hru13, Section 6.5]). In [Hem14] Hempel shows a finitary version of this result giving a certain "chain condition" for groups definable in n-dependent theories and demonstrating that every n-dependent field is Artin-Schreier closed. Some further questions and statements are mentioned in [She05,Section 5(H)].…”

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

On n-Dependence

Chernikov¹,

Palacín²,

Takeuchi³

2019

Notre Dame J. Formal Logic

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In this note we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence (for 1 ≤ n < ω) recently introduced by Shelah [She07]. In the same way as dependence of a theory means its inability to encode a bipartite random graph with a definable edge relation, n-dependence corresponds to the inability to encode a random (n + 1)-partite (n + 1)-hypergraph with a definable edge relation. Most importantly, we characterize n-dependence by counting ϕ-types over finite sets (generalizing Sauer-Shelah lemma and answering a question of Shelah from [She05]) and in terms of the collapse of random ordered (n + 1)-hypergraph indiscernibles down to order-indiscernibles (which implies that the failure of n-dependence is always witnessed by a formula in a single free variable).

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Section: A Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

On n-Dependence

Chernikov¹,

Palacín²,

Takeuchi³

2019

Notre Dame J. Formal Logic

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“…Observe that if F is Artin-Schreier closed, then the algebraic closure of the prime field of F is infinite in F . Thus, the above result holds for any infinite field of positive characteristic with finite burden and which in addition is NIP [10] or even n-dependent [7].…”

Section: Division Rings Of Finite Burdenmentioning

confidence: 64%

Division rings with ranks

Hempel

Palacín

2017

Proc. Amer. Math. Soc.

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Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden n has dimension at most n over its center, and any definable group of definable automorphisms of a field of burden n has size at most n. Additionally, interpretable division rings in o-minimal structures are shown to be algebraically closed, real closed or the quaternions over a real closed field.2000 Mathematics Subject Classification. 03C45, 03C60, 12E15.

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“…This result applies in particular to henselian valued NIP fields of positive characteristic that are not separably closed (see [JK15a, Corollary 3.18]), which subsequently was generalized from NIP to n-NIP in [Hem16,Proposition 7.4]. For the definition of NIP and more details on NIP fields as well as on related model-theoretic concepts, see [Sim15].…”

Section: Moreover [Jk16 Theorems a And B]mentioning

confidence: 99%