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Towards a finer classification of strongly minimal sets.
- Source :
-
Annals of Pure & Applied Logic . Feb2024, Vol. 175 Issue 2, pN.PAG-N.PAG. 1p. - Publication Year :
- 2024
-
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 | 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 Hrushovski's 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 = p n. 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *STEINER systems
*INDEPENDENT sets
*CLASSIFICATION
Subjects
Details
- Language :
- English
- ISSN :
- 01680072
- Volume :
- 175
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Annals of Pure & Applied Logic
- Publication Type :
- Academic Journal
- Accession number :
- 174013567
- Full Text :
- https://doi.org/10.1016/j.apal.2023.103376