Towards a finer classification of strongly minimal sets

Abstract

Let M be strongly minimal and constructed by a ‘Hrushovski construction’ with a single ternary relation. If the Hrushovski algebraization function μ is in a certain class $T$ (μ triples) we show that for independent I with $\left|I\right|>1$, ${\mathrm{dcl}}^{⁎}\left(I\right)=\varnothing$ (* means not in dcl of a proper subset). This implies the only definable truly n-ary functions f (f ‘depends’ on each argument), occur when $n=1$. We prove for Hrushovski's original construction and for the strongly minimal k-Steiner systems of Baldwin and Paolini that the symmetric definable closure, ${\mathrm{sdcl}}^{⁎}\left(I\right)=\varnothing$ (Definition 2.7). Thus, no such theory admits elimination of imaginaries. As, we show that in an arbitrary strongly minimal theory, elimination of imaginaries implies ${\mathrm{sdcl}}^{⁎}\left(I\right)\ne \varnothing$. 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 case structure depends on properties of the Hrushovski μ-function. The proofs depend on our introduction, for appropriate $G\subseteq \mathrm{aut}\left(M\right)$ (setwise or pointwise stabilizers of finite independent sets), the notion of a G-normal substructure $A$ of M and of a G-decomposition of any finite such $A$. These results lead to a finer classification of strongly minimal structures with flat geometry, according to what sorts of definable functions they admit.