On n-dependent groups and fields II
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On n-dependent groups and fields II

  • Author(s): Chernikov, Artem
  • Hempel, Nadja
  • et al.
Abstract

We continue the study of $n$-dependent groups, fields and related structures. We demonstrate that $n$-dependence is witnessed by formulas with all but one variable singletons, provide a type-counting criterion for $2$-dependence and use it to deduce $2$-dependence for compositions of NIP relations with arbitrary binary functions. We prove a result on intersections of type-definable connected components over generic sets of parameters in $n$-dependent groups, generalizing Shelah's results on absoluteness of $G^{00}$ in NIP theories and relative absoluteness of $G^{00}$ for $2$-dependent theories. We show that Granger's examples of non-degenerate bilinear forms over NIP fields are $2$-dependent, and characterize preservation of $n$-dependence under expansion by generic relations for geometric theories in terms of disintegration of their algebraic closure. Finally, we show that every infinite $n$-dependent valued field of positive characteristic is henselian, generalizing a recent result of Johnson for NIP.

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