Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Towards a Finer Classification of Strongly Minimal Sets   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.

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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
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The categories of this paper are: chem-ph
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The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ph
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th