Submission date: 5 December 2019

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.

Mathematics Subject Classification: 03C45, 03C60

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Full text arXiv 1912.02385: pdf, ps.