Resumen de Geometric structures with a dense independent subset
Alexander Berenstein, Evgueni Vassiliev
e study a generalization of the expansion by an independent dense set, introduced by Dolich, Miller, and Steinhorn in the o-minimal context, 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 TindTind , 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 TindTind is ωω -stable, supersimple, and superrosy, respectively, and its U-/SU-/thorn-rank is either 1 (if T is trivial) or ωω (if T is non-trivial). In the supersimple SU-rank 1 case, we obtain a description of forking and canonical bases in TindTind . We also show that if T is (strongly) dependent, then so is TindTind , and if T is non-trivial of finite dp-rank, then TindTind has dp-rank greater than n for every n<ωn<ω , but bounded by ωω . In the stable case, we also partially solve the question of whether any group definable in TindTind comes from a group definable in T.
