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Towards a Finer Classification of Strongly Minimal Sets

John T. Baldwin, Viktor V. Verbovskiy

Abstract

Let $M$ be strongly minimal and constructed by a `Hrushovski construction'. If the Hrushovski algebraization function $μ$ is in a certain class ${\mathcal T}$ ($μ$ 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 $μ$, 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.

Towards a Finer Classification of Strongly Minimal Sets

Abstract

Let be strongly minimal and constructed by a `Hrushovski construction'. If the Hrushovski algebraization function is in a certain class ( triples) we show that for independent with , (* means not in of a proper subset). This implies the only definable truly -ary function ( `depends' on each argument), occur when . We prove, indicating the dependence on , for Hrushovski's original construction and including analogous results for the strongly minimal -Steiner systems of Baldwin and Paolini 2021 that the symmetric definable closure, , 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 . The proofs depend on our introduction for appropriate the notion of a -normal substructure of and of a -decomposition of . These results lead to a finer classification of strongly minimal structures with flat geometry; according to what sorts of definable functions they admit.

Paper Structure

This paper contains 12 sections, 48 theorems, 60 equations, 10 figures.

Table of Contents

  1. Two Contexts
  2. General Context
  3. $3$-hypergraphs and Linear Spaces
  4. Elimination of imaginaries and (Symmetric) Definable closure
  5. $G$-Decomposition
  6. $\mathop{\mathrm{dcl}}\nolimits$ in Hrushovski's first example $\hat{T}_\mu$
  7. $G_I$: No truly $n$-ary definable functions
  8. Counterexample
  9. Bouquets and Flowers
  10. $G_{\{ I\}}$: elimination of imaginaries fails
  11. Steiner Systems
  12. Further work

Key Result

Theorem \oldthetheorem

Let $T_\mu$ be a strongly minimal theory as in Notation defT. Let $I =\{a_1,\dots, a_v\}$ be a tuple of independent points with $v\ge 2$. Consequently, in both cases $T_\mu$ does not admit elimination of imaginaries.Note that each of the conditions $\mathop{\mathrm{dcl}}\nolimits^*(I) = \emptyset$ and ${\rm sdcl}^*(I) =\emptyset$ is stronger than failure to eliminate of imaginaries. For example,

Figures (10)

  • Figure 1: From a linear to a tree-decomposition: One Step
  • Figure 2: Chain
  • Figure 3: Decomposition
  • Figure 4: $A^2_{2,2} = \mathfrak A^0$
  • Figure 5: Illustrating Notation \ref{['pm']}
  • ...and 5 more figures

Theorems & Definitions (147)

  • Theorem \oldthetheorem: Main Results
  • Remark \oldthetheorem: Classes of Theories with flat ${\rm acl}$-geometries
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem: Context
  • ...and 137 more