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Strongly minimal Steiner systems I: Existence

John Baldwin & Gianluca Paolini

Journal of Symbolic Logic 86 (4):1486-1507 (2021)

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Stable generic structures.John T. Baldwin & Niandong Shi - 1996 - Annals of Pure and Applied Logic 79 (1):1-35.details Hrushovski originated the study of “flat” stable structures in constructing a new strongly minimal set and a stable 0-categorical pseudoplane. We exhibit a set of axioms which for collections of finite structure with dimension function δ give rise to stable generic models. In addition to the Hrushovski examples, this formalization includes Baldwin's almost strongly minimal non-Desarguesian projective plane and several others. We develop the new case where finite sets may have infinite closures with respect to the dimension function δ. In (...) Download Export citation Bookmark 39 citations
On Model-Completeness.Per Lindström - 1964 - Theoria 30 (3):183-196.details Download Export citation Bookmark 19 citations
Model theory of Steiner triple systems.Silvia Barbina & Enrique Casanovas - 2020 - Journal of Mathematical Logic 20 (2):2050010.details A Steiner triple system (STS) is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite STS has a Fraïssé limit M_F. Here, we show that the theory T of M_F is the model completion of the theory of STSs. We also prove that T is not small and it has quantifier elimination, (...) Download Export citation Bookmark 3 citations
An exposition of Hrushovskiʼs New Strongly Minimal Set.Martin Ziegler - 2013 - Annals of Pure and Applied Logic 164 (12):1507-1519.details We give an exposition of Hrushovskiʼs New Strongly Minimal Set : A strongly minimal theory which is not locally modular but does not interpret an infinite field. We give an exposition of his construction. Download Export citation Bookmark 3 citations
A new strongly minimal set.Ehud Hrushovski - 1993 - Annals of Pure and Applied Logic 62 (2):147-166.details We construct a new class of 1 categorical structures, disproving Zilber's conjecture, and study some of their properties. Download Export citation Bookmark 89 citations
The geometry of Hrushovski constructions, I: The uncollapsed case.David M. Evans & Marco S. Ferreira - 2011 - Annals of Pure and Applied Logic 162 (6):474-488.details An intermediate stage in Hrushovski’s construction of flat strongly minimal structures in a relational language L produces ω-stable structures of rank ω. We analyze the pregeometries given by forking on the regular type of rank ω in these structures. We show that varying L can affect the isomorphism type of the pregeometry, but not its finite subpregeometries. A sequel will compare these to the pregeometries of the strongly minimal structures. Download Export citation Bookmark 3 citations
Strongly minimal Steiner systems I: Existence.John Baldwin & Gianluca Paolini - 2021 - Journal of Symbolic Logic 86 (4):1486-1507.details A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner k-system is a linear space such that each line has size exactly k. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a vocabulary $\tau $ with a single ternary relation R. We prove that for every integer k there exist $2^{\aleph _0}$ -many integer valued functions $\mu $ such that (...) Download Export citation Bookmark 2 citations

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