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21
Ample dividing
 J. Symbolic Logic
"... We construct a stable onebased, trivial theory with a reduct which is not trivial. This answers a question of John B. Goode. Using this, we construct a stable theory which is nample for all natural numbers n, and does not interpret an infinite group. ..."
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We construct a stable onebased, trivial theory with a reduct which is not trivial. This answers a question of John B. Goode. Using this, we construct a stable theory which is nample for all natural numbers n, and does not interpret an infinite group.
Die böse Farbe
, 2007
"... We construct a bad field in characteristic zero. That is, we construct an algebraically closed field which carries a notion of dimension analogous to Zariskidimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a ..."
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We construct a bad field in characteristic zero. That is, we construct an algebraically closed field which carries a notion of dimension analogous to Zariskidimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a longstanding open question by Zilber.
Trivial stable structures with nontrivial reducts
 J. London Math. Soc
, 2003
"... Abstract. We offer a new viewpoint on some of the generic structures constructed using Hrushovski’s predimensions and show that they are natural reducts of quite straightforward trivial, onebased stable structures. 2000 Mathematics Subject Classification: 03C45. ..."
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Abstract. We offer a new viewpoint on some of the generic structures constructed using Hrushovski’s predimensions and show that they are natural reducts of quite straightforward trivial, onebased stable structures. 2000 Mathematics Subject Classification: 03C45.
The geometries of the Hrushovski constructions
, 2009
"... In 1984 Zilber conjectured that any strongly minimal structure is geometrically equivalent to one of the following types of strongly minimal structures, in the appropriate language: Pure sets, Vector Spaces over a fixed Division Ring and Algebraically Closed Fields. In 1993, in his article ‘A new st ..."
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Cited by 4 (2 self)
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In 1984 Zilber conjectured that any strongly minimal structure is geometrically equivalent to one of the following types of strongly minimal structures, in the appropriate language: Pure sets, Vector Spaces over a fixed Division Ring and Algebraically Closed Fields. In 1993, in his article ‘A new strongly minimal set’ Hrushovski produced a family of counterexamples to Zilber’s conjecture. His method consists in two steps. Firstly he builds a ‘limit’ structure from a suitable class of finite structures in a language consisting only of a ternary relational symbol. Secondly, in a step called the collapse, he defines a continuum of subclasses such that the corresponding ‘limit’ structures are new strongly minimal structures. These new strongly minimal structures are non isomorphic but Hrushovski then asks if they are geometrically equivalent. We first analyze the pregeometries arising from different variations of the construction before the collapse. In particular we prove that if we repeat the construction starting with an nary relational symbol instead of a 3ary relational symbol, then the pregeometries associated to the corresponding ‘limit’ structures are not locally isomorphic when we vary the arity. Second we prove that these new strongly minimal structures are geometrically equivalent. In fact we prove that their geometries are isomorphic to the geometry of the ‘limit’ structure obtained before the collapse.
The geometry of Hrushovski constructions II. The strongly minimal case
"... We investigate the isomorphism types of combinatorial geometries arising from Hrushovski’s flat strongly minimal structures and answer some questions from Hrushovski’s original paper. ..."
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We investigate the isomorphism types of combinatorial geometries arising from Hrushovski’s flat strongly minimal structures and answer some questions from Hrushovski’s original paper.
Some remarks on generic structures
, 2008
"... We show that the ℵ0categorical structures produced by Hrushovski’s predimension construction with a control function fit neatly into Shelah’s SOPn hierarchy: if they are not simple, then they have SOP3 and NSOP4. We also show that structures produced without using a control function can be undeci ..."
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We show that the ℵ0categorical structures produced by Hrushovski’s predimension construction with a control function fit neatly into Shelah’s SOPn hierarchy: if they are not simple, then they have SOP3 and NSOP4. We also show that structures produced without using a control function can be undecidable and have SOP.