On model theory of covers of algebraically closed fields

TL;DR

This paper investigates covers of algebraically closed fields as quasiminimal pregeometry structures, demonstrating they meet Zariski-like axioms that extend Zariski geometry concepts into a broader, non-elementary framework.

Contribution

It establishes that covers of algebraically closed fields satisfy Zariski-like axioms, broadening the understanding of their model-theoretic properties beyond classical Zariski geometries.

Findings

Covers satisfy axioms for Zariski-like structures.

Irreducible sets generalize properties of Zariski closed sets.

Some compactness traces are preserved in the non-elementary setting.

Abstract

We study covers of the multiplicative group of an algebraically closed field as quasiminimal pregeometry structures and prove that they satisfy the axioms for Zariski-like structures presented in \cite{lisuriart}, section 4. These axioms are intended to generalize the concept of a Zariski geometry into a non-elementary context. In the axiomatization, it is required that for a structure $\M$, there is, for each $n$, a collection of subsets of $\M^n$, that we call the \emph{irreducible sets}, satisfying certain properties. These conditions are generalizations of some qualities of irreducible closed sets in the Zariski geometry context. They state that some basic properties of closed sets (in the Zariski geometry context) are satisfied and that specializations behave nicely enough. They also ensure that there are some traces of Compactness even though we are working in a non-elementary context.