We study expansions of a vector space V over a field F, possibly with extra structure, with a generic submodule over a subring of F. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP1, NTP2 and NSOP1. We also study induced independence relations in the expansion.

Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-basedω-categorical geometric theory interprets an infinite vector space over a finite field.

Abstract The weak non-finite cover property (wnfcp) was introduced in [1] in connection with “axiomatizability” of lovely pairs of models of a simple theory. We find a combinatorial condition on a simple theory equivalent to the wnfcp, yielding a direct proof that the non-finite cover property implies the wnfcp, and that the wnfcp is preserved under reducts. We also study the question whether the wnfcp is preserved when passing from a simple theory T to the theory Tp of lovely pairs of models of T (true in the stable case). While the question remains open, we show, among other things, that if (for a T with the wnfcp) Tp is low, then TP has the wnfcp. To study this question, we describe “double lovely pairs”, and, along the way, we develop the notion of a “lovely n-tuple” of models of a simple theory, which is an analogue of the notion of a beautiful tuple of models of stable theories [2].