Finding a field in a Zariski-like structure

TL;DR

This paper proves that certain Zariski-like structures without non-classical groups interpret algebraically closed fields when their associated pregeometry is non locally modular.

Contribution

It establishes a new criterion linking the non local modularity of the pregeometry to the interpretation of algebraically closed fields in Zariski-like structures.

Findings

Non classical groups are not interpreted in the structure

Non local modularity implies the interpretation of an algebraically closed field

Provides conditions under which Zariski-like structures interpret fields

Abstract

We show that if $\M$ is a Zariski-like structure (see \cite{lisuriart}) that does not interpret a non-classical group, and the canonical pregeometry obtained from the bounded closure operator (bcl) is non locally modular, then $\M$ interprets an algebraically closed field.