Indifference to symmetry in Hrushovski's ab initio construction

TL;DR

This paper investigates the relationship between symmetric and non-symmetric Hrushovski constructions, showing they are mutually reducts and share isomorphic pregeometries, revealing an unexpected indifference to symmetry.

Contribution

It demonstrates that symmetric and non-symmetric Hrushovski constructions are closely related, with each being a reduct of the other, and their associated pregeometries are isomorphic.

Findings

$ ext{M}_{ ot ext{sim}}$ is a proper reduct of $ ext{M}_{ ext{sim}}$

The combinatorial pregeometries of both structures are isomorphic

Symmetry does not fundamentally alter the combinatorial geometry

Abstract

Denote Hrushovski's non-collapsed ab initio construction for an $n$-ary relation by $\mathcal{M}_{\not\sim}$ and the analogous construction for a symmetric $n$-ary relation by $\mathcal{M}_{\sim}$. We show that $\mathcal{M}_{\not\sim}$ is isomorphic to a proper reduct of $\mathcal{M}_{\sim}$ and vice versa, and that the combinatorial pregeometries associated with both structures are isomorphic.