On reducts of Hrushovski's construction - the non-collapsed case

TL;DR

This paper investigates the reducts of a non-collapsed Hrushovski structure, showing it has a proper reduct with a unique regular type and isomorphic geometry, raising questions about bi-interpretability and further reducts.

Contribution

It demonstrates that the non-collapsed Hrushovski structure admits a proper reduct that is a Fraïssé-Hrushovski limit with preserved geometric properties.

Findings

The reduct has a unique regular type of rank ω.

Its geometry is isomorphic to the original structure's generic type.

The reduct is the Fraïssé-Hrushovski limit of its own age.

Abstract

We show that the rank {\omega} structure obtained by the non-collapsed version of Hrushovski's amalgamation construction has a proper reduct. We show that this reduct is the Fra\"iss\'e-Hrushovski limit of its own age with respect to a pre-dimension function generalising Hrushovski's pre-dimension function. It follows that this reduct has a unique regular type of rank {\omega}, and we prove that its geometry is isomorphic to the geometry of the generic type in the original structure. We ask whether our reduct is bi-interpretable with the original structure and whether it, too, has proper reducts with the same geometry.