Generic Expansions by a Reduct

TL;DR

This paper explores how expanding a theory by adding predicates for submodels of a reduct can preserve or alter properties like stability and simplicity, providing new examples of theories with specific model-theoretic features.

Contribution

It introduces a framework for such expansions that admits a model companion and analyzes the transfer of model-theoretic properties, including concrete new examples.

Findings

Expansion admits a model companion under certain conditions

Preservation of NSOP1, simplicity, and stability is characterized

New NSOP1 theories that are not simple are constructed

Abstract

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 the nice features of the theory $T$ transfer to $TS$. In particular, we study conditions for which this expansion preserves the $\NSOP{1}$-ness, the simplicity or the stability of the starting theory $T$. We give concrete examples of new $\NSOP{1}$ 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 \emph{any} characteristic by a generic multiplicative subgroup.