Towards a finer classification of strongly minimal sets
Baldwin J.T. Verbovskiy V.V.
February 2024Elsevier B.V.
Annals of Pure and Applied Logic
2024#175Issue 2
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 |I|>1, dcl⁎(I)=∅ (* 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 Hrushovskis original construction and for the strongly minimal k-Steiner systems of Baldwin and Paolini that the symmetric definable closure, sdcl⁎(I)=∅ (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 sdcl⁎(I)≠∅. 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=pn. The case structure depends on properties of the Hrushovski μ-function. The proofs depend on our introduction, for appropriate G⊆aut(M) (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.
Group actions on homogeneous structures , Hrushovski construction , Steiner systems , Strongly minimal sets , Zilber conjecture
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University of Illinois at Chicago, United States
Satbayev University, Kazakhstan
University of Illinois at Chicago
Satbayev University
10 лет помогаем публиковать статьи Международный издатель
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