Generic expansion and Skolemization in NSOP1 theories

We study expansions of NSOP1 theories that preserve NSOP1. We prove that if T is a model complete NSOP1 theory eliminating the quantifier there exists(infinity), then the generic expansion of T by arbitrary constant, function, and relation symbols is still NSOP1. 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 NSOP1 theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in NSOP1 theories, adding instances of algebraic independence to their conclusions. (C) 2018 Elsevier B.V. All rights reserved.