Neostability transfers in derivation-like theories

TL;DR

This paper introduces derivation-like theories relative to a base theory, demonstrating that key model-theoretic properties transfer from the base to its model companion, with applications beyond differential fields.

Contribution

It formalizes the notion of derivation-like theories and proves the transfer of important properties to their model companions, expanding understanding of their structural behavior.

Findings

Model properties transfer from $T_0$ to $T_+$

Examples of derivation-like theories are abundant

Applicable beyond differential fields

Abstract

Motivated by structural properties of differential field extensions, we introduce the notion of a theory $T$ being derivation-like with respect to another model complete theory $T_0$. We prove that when $T$ admits a model companion $T_+$, several model-theoretic properties transfer from $T_0$ to $T_+$. These properties include completeness, quantifier elimination, stability, simplicity, and NSOP$_1$. We also observe that, aside from the theory of differential fields, examples of derivation-like theories are plentiful.