Generic expansions by a reduct

摘要

Consider the expansion T-S of a theory T by a predicate for a submodel of a reduct T-0 of T. We present a setup in which this expansion admits a model companion TS. We show that some of the nice features of the theory T transfer to T-S. In particular, we study conditions for which this expansion preserves the NSOP1-ness, the simplicity or the stability of the starting theory T. We give concrete examples of new NSOP1 not simple theories obtained by this process, among them the expansion of a perfect omega-free PAC field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of any characteristic by a generic multiplicative subgroup.