Generic trivializations of geometric theories

Abstract

Abstract. We study the theory T * of the structure induced by parameter free formulas on a dense algebraically independent subset of a model of a geometric theory T . We show that while being a trivial geometric theory, T * inherits most of the model theoretic complexity of T related to stability, simplicity, rosiness, N IP and N T P 2 . In particular, we show that T is strongly minimal, supersimple of SU-rank 1, or NIP exactly when so is T * . We show that if T is superrosy of thorn rank 1, then so is T * , and that the converse holds if T satis es acl = dcl.

Definable groups in dense pairs of geometric structures

“…The condition of TEA/WTEA allows us to describe new definable sets in terms of old definable sets up to small sets. A version of the following result for sets of dimension one was central to many arguments on dense/codense pairs [3,5,6], in particular for understanding which model-theoretic properties transfer form T h(M) to T h(M, P). Proposition 2.6 Let (M, P) be a dense/codense-structure satisfying WTEA, let Z ⊂ M n be L-definable and let Y ⊂ Z be L P -definable.…”

“…In Sect. 6, we move into the setting of pairs (R, G) where R is a real closed field and G is a subgroup of R >0 or the unit circle S(R) with the Mann property. When G is interpreted as a subgroup of R >0 , we analyze the definable subgroups of G in the language of pairs and show that they are already definable in the pure ordered group language.…”

H-structures and generalized measurable structures

“…Let us consider the special case of measures of L H -definable subsets of H k with dimension k. By Lemma 2.7 a definable subset Y of H(M ) k is the set of solutions of a formula of the form (x ∈ H k ) ∧ θ(x, c), where c is a tuple from M and θ(x, ȳ) is an L-formula. Recall that H with the induced structure from M is a generic trivialization of M (see [4]) and that there is a nice correspondence between its theory T * and the theory T . For example T is supersimple of SU -rank 1 or strongly minimal if and only if T * is supersimple of SU -rank 1 or strongly minimal (see also [4]).…”

“…Recall that H with the induced structure from M is a generic trivialization of M (see [4]) and that there is a nice correspondence between its theory T * and the theory T . For example T is supersimple of SU -rank 1 or strongly minimal if and only if T * is supersimple of SU -rank 1 or strongly minimal (see also [4]). If Y is a definable subset of H(M ) k of dimension k, the density property implies that generic properties of θ(x, c) hold for Y .…”

A. D. Taimanov and model theory in Kazakhstan

“…(Pillay-Vassiliev, 2004), A. Berenstein and E. Vassiliev considered expansions of geometric theories by a unary predicate with density and extension properties and studied connection between properties of the original theory with properties of such expansions[48,49,50] (2010-2019).I 6 -weakly o-minimal theories [Macpherson-Marker-Steinhorn, Baizhanov, Verbovskiy, Kulpeshov, Aref'ev].…”