Existentially closed models of fields with a distinguished submodule

TL;DR

This paper investigates the model theory of fields with a distinguished submodule, showing that in positive characteristic, the class of existentially closed models is elementary and analyzing its classification in Robinson's logic.

Contribution

It extends the study of existentially closed models of fields with a submodule to Robinson's logic, demonstrating NSOP$_1$ and TP$_2$ properties.

Findings

The class is elementary in positive characteristic.

The class is NSOP$_1$ and TP$_2$ in Robinson's logic.

Provides new insights into model-theoretic classification of these models.

Abstract

This paper deals with the class of existentially closed models of fields with a distinguished submodule (over a fixed subring). In the positive characteristic case, this class is elementary and was investigated by the first-named author. Here we study this class in Robinson's logic, meaning the category of existentially closed models with embeddings following Haykazyan and Kirby, and prove that in this context this class is NSOP$_1$ and TP$_2$.