The generic flat pregeometry

TL;DR

This paper analyzes the structure of flat pregeometries arising from Hrushovski constructions, establishing their properties such as saturation, stability, and quantifier-elimination, and showing their relation to bounded-arity cases.

Contribution

It demonstrates that flat pregeometries form an amalgamation class with a generic that is saturated, stable, and admits quantifier-elimination, linking unbounded and bounded arity constructions.

Findings

The generic flat pregeometry is saturated and $ ext{ω}$-stable.

The theory admits quantifier-elimination to boolean combinations of $ ext{∃∀}$-formulas.

Pregeometries of bounded-arity constructions form an elementary chain.

Abstract

We examine the first order structure of pregeometries of structures built via Hrushovski constructions. In particular, we show that the class of flat pregeometries is an amalgamation class such that the pregeometry of the unbounded arity Hrushovski construction is precisely its generic. We show that the generic is saturated, provide an axiomatization for its theory, show that the theory is $\omega$-stable, and has quantifier-elimination down to boolean combinations of $\exists\forall$-formulas. We show that the pregeometries of the bounded-arity Hrushovski constructions satisfy the same theory, and that they in fact form an elementary chain.