# Generic expansion and Skolemization in NSOP$_1$ theories

## TL;DR

This paper investigates how to expand NSOP$_1$ theories while preserving their properties, demonstrating that generic expansions and Skolemizations maintain NSOP$_1$ under certain conditions, with detailed analysis and strengthened independence properties.

## Contribution

It proves that NSOP$_1$ theories can be generically expanded with constants, functions, and relations, and also with Skolem functions, preserving NSOP$_1$ under specific assumptions.

## Key findings

- Generic expansions preserve NSOP$_1$ in certain theories.
- Skolemization can be performed while maintaining NSOP$_1$.
- Strengthened properties of Kim-independence are established.

## 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.