Submission date: 15 November 2013

Abstract:

We define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is d^p-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces. We introduce a way to measure the dependence of a tuple a from a set B over another set A. We prove basic properties of the notion, e.g. that a is independent of B over A (in the usual sense of homogeneous model theory) iff the measure of dependence is <ε for all ε >0. As an application, we show that weak simplicity implies (a very strong form of) simplicity and study the question when the dependence inside a set of all realisations of some type can be seen to arise from a pregeometry in cases when the type is not regular. In the end of the paper, we demonstrate our notions and results in an example class built from the p-adic integers.

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Full text arXiv 1311.3806: pdf, ps.