Journal of Symbolic Logic

Abelian Groups with Modular Generic

James Loveys

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Let $G$ be a stable abelian group with regular modular generic. We show that either 1. there is a definable nongeneric $K \leq G$ 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 $G \times G$ realizing distinct strong types (when regarded as elements of $G^{eq}$). In the latter case one can choose almost 0-definable subgroups representing the elements of the division ring. We find a bound $((G : G^0))$ 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 $\mathrm{Th}(G/H)$ is independent of the particular model $G$ as long as $G/H$ is infinite.

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J. Symbolic Logic, Volume 56, Issue 1 (1991), 250-259.

First available in Project Euclid: 6 July 2007

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Loveys, James. Abelian Groups with Modular Generic. J. Symbolic Logic 56 (1991), no. 1, 250--259.

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