On the Theory of Specialisations of Regular Covers of Zariski Structures

TL;DR

This paper explores the extension and uniqueness of specialisations in regular covers of Zariski structures, revealing structural and topological properties essential for a robust theory of specialisations.

Contribution

It investigates the existence and uniqueness of specialisation extensions in Zariski structures and identifies conditions for a 'good' theory of specialisations.

Findings

Identified structural properties of regular covers of Zariski structures.

Determined topological conditions for a 'good' specialisation theory.

Defined a subclass of Zariski structures with desirable specialisation properties.

Abstract

In algebraic geometry specialisations and valuations play and important role. In this paper we start investigating analogous structures for Zariski structures. Specifically, we look into the existence and uniqueness properties of extensions of universal specialisations from a base Zariski structure to its regular cover. In the process we begin to uncover some structural properties of regular covers of Zariski structures, and also to uncover the type of topological properties necessary for a Zariski structure to have a "good" theory of specialisations. A subclass of Zariski structures is identified with a ``good'' theory of specialisations.