Geometric structures with a dense independent subset

Abstract

Abstract. We generalize the work of [13] 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 SUrank 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. In the stable case, we also partially solve the question of whether any group definable in T ind comes from a group definable in T .

Dense codense predicates and the NTP2

“…We start by recalling some definitions and results presented in . We also refine some of their results on definable sets (Propositions and below).…”

“…In the remainder of this section, we prove Propositions and which are refined versions of the original theorems by Berenstein and Vassiliev . (The original versions do not make explicit references to parameters.)…”

“…In this section, we review the notions of H ‐ structures and lovely pairs of geometric theories developed by Berenstein and Vassiliev .…”

“…Not every model of may be an H ‐structure (or a lovely pair), but all the sufficiently saturated ones are (cf. [, Examples 2.11 & 2.12]).…”

Supersimple structures with a dense independent subset

“… Abstract Based on the work done in in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking‐independent elements that is dense inside a partial type , which we call H ‐structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again H ‐structures.…”