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Abelian groups with modular generic

Published online by Cambridge University Press:  12 March 2014

James Loveys
James Loveys*
Affiliation:
Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 2K6, Canada
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Abstract

Let G be a stable abelian group with regular modular generic. We show that either

1. there is a definable nongeneric K ≤ G such that G/K has definable connected component and so strongly regular generics, or

2. distinct elements of the division ring yielding the dependence relation are represented by subgroups of G × G realizing distinct strong types (when regarded as elements of Geq).

In the latter case one can choose almost 0-definable subgroups representing the elements of the division ring. We find a bound ((G: G0)) for the size of the division ring in case G has no definable subgroup K so that G/K is infinite with definable connected component. We show in case (2) that the group G/H, where H consists of all nongeneric points of G, inherits a weakly minimal group structure from G naturally, and Th(G/H) is independent of the particular model G as long as G/H is infinite.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 1 , March 1991 , pp. 250 - 259
Copyright
Copyright © Association for Symbolic Logic 1991

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References

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