Generic expansion and Skolemization in NSOP$_1$ theories

Abstract

We study expansions of NSOP$_1$ theories that preserve NSOP$_1$. We prove that if $T$ is a model complete NSOP$_1$ theory eliminating the quantifier $\exists^{\infty}$, then the generic expansion of $T$ by arbitrary constant, function, and relation symbols is still NSOP$_1$. We give a detailed analysis of the special case of the theory of the generic $L$-structure, the model companion of the empty theory in an arbitrary language $L$. Under the same hypotheses, we show that $T$ may be generically expanded to an NSOP$_1$ theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in NSOP$_1$ theories, adding instances of algebraic independence to their conclusions.