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Preprint Number 490

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490. Alexander Berenstein and Evgueni Vassiliev
Geometric structures with a dense independent subset
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Submission date: 7 August 2012.

Abstract:

We generalize the work of Dolich, Miller and Steinhorn on expansions of o-minimal structures with dense independent subsets, to the setting of geometric structures. We introduce the notion of an H-structure of a geometric theory T, show that H-structures exist and are elementarily equivalent, and establish some basic properties of the resulting complete theory T^{ind}, including quantifier elimination down to “H-bounded” formulas, and a description of definable sets and algebraic closure. We show that if T is strongly minimal, supersimple of SU-rank 1, or superrosy of thorn rank 1, then T^{ind} is ω-stable, supersimple, and superrosy, respectively, and its U-/SU-/thorn rank is either 1 (if T is trivial) or ω (if T is non-trivial). In the supersimple SU-rank 1 case, we obtain a description of forking and canonical bases in T^{ind}. We also show that if T is (strongly) dependent, then so is T^{ind}, and if T is non-trivial of finite dp-rank, then T^{ind} has dp-rank greater than n for every n<ω, but bounded by ω. We also partially solve the question of whether any group definable in T^{ind} comes from a group interpretable in T.

Mathematics Subject Classification: 03C10, 03C45, 03C64

Keywords and phrases: geometric structures, strongly minimal theories, U-rank one theories, rosy theories, strongly dependent theories, definable groups.

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