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MEASURING DEPENDENCE IN METRIC ABSTRACT ELEMENTARY CLASSES WITH PERTURBATIONS

Published online by Cambridge University Press:  09 January 2018

ÅSA HIRVONEN  and
TAPANI HYTTINEN
ÅSA HIRVONEN
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF HELSINKI P.O. BOX 68, 00014HELSINKI FINLANDE-mail:asa.hirvonen@helsinki.fi
TAPANI HYTTINEN
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF HELSINKI P.O. BOX 68, 00014HELSINKI FINLANDE-mail:tapani.hyttinen@helsinki.fi
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Abstract

We define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is dp-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces and we give a new example of such a class based on B. Zilber’s approximations of Weyl algebras. 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 if and only if the measure of dependence is < ε for all ε > 0. In well behaved situations, the measure corresponds to the distance to a free extension. As an example of our measure of dependence we show a connection between the measure and entropy in models from quantum mechanics in which the spectrum of the observable is discrete. As an application, we show that weak simplicity implies a very strong form of simplicity and study the question of 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 one more example: a class built from the p-adic integers.

Keywords

Primary 03C45Secondary 3C52metric structuresperturbationindependence
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1199 - 1228
Copyright
Copyright © The Association for Symbolic Logic 2017 

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