Generalised Stability of ultraproducts of finite residue rings

TL;DR

This paper investigates the model-theoretic properties of ultraproducts of finite residue rings, identifying conditions on ultrafilters that determine whether these structures are simple, NIP, NTP2, or TP2, with all cases occurring.

Contribution

It provides new criteria linking ultrafilter properties to the stability and classification of ultraproducts of finite residue rings.

Findings

Ultraproducts can be simple, NIP, NTP2, or TP2 depending on ultrafilter conditions.

All four model-theoretic cases are demonstrated to occur.

Conditions on ultrafilters are characterized for each case.

Abstract

We study ultraproducts of finite residue rings $\underset{n\in\mathbb{N}}{\prod}\mathbb{Z}/n\mathbb{Z} \diagup \mathcal{U} $ where $\mathcal{U}$ is a non-principal ultrafilter. We find sufficient conditions of the ultrafilter $\mathcal{U}$ to determine if the resulting ultraproduct $\underset{n\in\mathbb{N}}{\prod}\mathbb{Z}/n\mathbb{Z} \diagup \mathcal{U}$ has simple, NIP, NTP2 but not simple nor NIP, or TP2 theory, noting that all these four cases occur.