Bilinear spaces over a fixed field are simple unstable

TL;DR

This paper explores the model theory of bilinear spaces over fixed fields, establishing their simplicity and instability, and characterizing their existentially closed models using category-theoretic and positive logic frameworks.

Contribution

It introduces a novel analysis of bilinear spaces over infinite fields using positive logic, demonstrating their simplicity and instability, and characterizing their existentially closed models.

Findings

Linear independence forms a simple unstable independence relation.

Bilinear spaces over countable fields are ω-categorical.

The theory exhibits dividing with local character but many types.

Abstract

We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can be, and has been, done in the classical framework of full first-order logic. For infinite fields we need different logical frameworks. First we take a category-theoretic approach, which requires very little set-up. We show that linear independence forms a simple unstable independence relation. With some more work we then show that we can also work in the framework of positive logic, which is much more powerful than the category-theoretic approach and much closer to the classical framework of full first-order logic. We fully characterise the existentially closed models of the arising positive theory. Using the independence relation from before we conclude that the theory is simple unstable, in the sense that dividing has local character but there are many distinct types. We also provide positive version of what is commonly known as the Ryll-Nardzewski theorem for $\omega$-categorical theories in full first-order logic, from which we conclude that bilinear spaces over a countable field are $\omega$-categorical.