An ab initio construction of a geometry

TL;DR

This paper demonstrates that the geometry derived from Hrushovski's ab initio construction for a specific n-ary relation can itself be constructed using a similar Hrushovski method, revealing a recursive structural property.

Contribution

It shows that the geometry associated with Hrushovski's ab initio construction can be reconstructed as a Hrushovski construction itself, extending understanding of its structural properties.

Findings

The geometry of the construction can be obtained via a Hrushovski construction.

The approach applies to a single n-ary relation with certain dependence restrictions.

The result links the geometry directly to the original Hrushovski framework.

Abstract

We show that the geometry of Hrushovski's ab initio construction for a single $n$-ary relation not-permitting dependent sets of size less than $n$, when restricted to $n$-tuples, can be itself constructed as a Hrushovski construction.