Towards a Finer Classification of Strongly Minimal Sets

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 $|I|>1$, ${\mathrm{dcl}}^{\ast }(I)=\varnothing$ (* means not in $\mathrm{dcl}$ 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, ${\mathrm{sdcl}}^{\ast }(I)=\varnothing$, 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 $G\subseteq \mathrm{aut}(M)$ 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.

58 pages, 8 figures; The revision modifies the treatment of weak elimination of imaginaries on page 10; it has no effect on the main arguments and results

References in corpus (5)

• The geometry of Hrushovski constructions, I. The uncollapsed case
• Model theory of Steiner triple systems
• Countable homogeneous Steiner triple systems avoiding specified subsystems
• Strongly minimal Steiner Systems III: Path graphs and sparse configurations
• The generic flat pregeometry