Towards a Finer Classification of Strongly Minimal Sets

TL;DR

This paper investigates the structure of strongly minimal sets constructed via Hrushovski methods, revealing limitations on definable functions and implications for their geometric classification.

Contribution

It introduces the notion of G-normal substructures and G-decompositions to analyze definable functions, providing a finer classification of strongly minimal structures with flat geometry.

Findings

Only trivial definable functions exist for independent tuples when μ is in class T.

Strongly minimal Steiner systems with line-length ≥ 4 do not interpret a quasigroup.

The theory does not admit elimination of imaginaries for these structures.

Abstract

Let $M$ be strongly minimal and constructed by a `Hrushovski construction'. If the Hrushovski algebraization function $\mu$ is in a certain class ${\mathcal T}$ ($\mu$ triples) we show that for independent $I$ with $|I| >1$, ${\rm dcl}^*(I)= \emptyset$ (* means not in ${\rm dcl}$ of a proper subset). This implies the only definable truly $n$-ary function $f$ ($f$ `depends' on each argument), occur when $n=1$. We prove, indicating the dependence on $\mu$, for Hrushovski's original construction and including analogous results for the strongly minimal $k$-Steiner systems of Baldwin and Paolini 2021 that the symmetric definable closure, ${\rm sdcl}^*(I) =\emptyset$, and thus the theory does not admit elimination of imaginaries. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if $k = p^n$. The proofs depend on our introduction for appropriate $G \subseteq {\rm aut}(M)$ the notion of a $G$-normal substructure ${\mathcal A}$ of $M$ and of a $G$-decomposition of ${\mathcal A}$. These results lead to a finer classification of strongly minimal structures with flat geometry; according to what sorts of definable functions they admit.