Generic derivations on algebraically bounded structures

TL;DR

This paper investigates the model-theoretic properties of algebraically bounded structures with derivations, establishing the existence of model completions and stability/NIP transfer for theories with multiple derivations.

Contribution

It proves the existence of model completions for theories of algebraically bounded structures with derivations and shows stability/NIP preservation for multiple derivations.

Findings

Model completion exists for $T^{ ext{δ}}$ when T is model complete.

Stability/NIP are preserved in the model completion for theories with derivations.

Results extend to multiple, commuting or non-commuting derivations.

Abstract

Let K be an algebraically bounded structure and T be its theory. If T is model complete, then the theory of K endowed with a derivation, denoted by $T^{\delta}$, has a model completion. Additionally, we prove that if the theory T is stable/NIP then the model completion of $T^{\delta}$ is also stable/NIP. Similar results hold for the theory with several derivations, either commuting or non-commuting.