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تسجيل مستخدم جديدTowards a Finer Classification of Strongly Minimal Sets
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تأليف
John T. Baldwin
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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 Hrushovskis 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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