@MISC{Hasson_dmpin, author = {Assaf Hasson and Ehud Hrushivski}, title = {DMP IN STRONGLY MINIMAL SETS}, year = {} }

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Abstract

Abstract. We construct a strongly minimal set which is not a finite cover of one with DMP. We also prove that for a strongly minimal theory T generic automorphisms exist iff T has DMP, thus proving a conjecture of Kikyo and Pillay. Recall that a strongly minimal theory T has the Definable Multiplicity Property (DMP) if for all natural k, m and ϕ(¯x, ¯ b) of rank k, multiplicity m there exists a formula θ ∈ tp ( ¯ b) such that for all ¯ b ′ | = θ, rk{ϕ(¯x, ¯ b ′)} = k and mult{ϕ(¯x, ¯ b ′)} = m. This definition was introduced in [2], where it was asked whether every strongly minimal set is a finite cover of one with DMP, i.e. whether for every strongly minimal T there exists a definable equivalence relation E with finite classes such that T/E has DMP. In [4] Kikyo and Pillay prove that a strongly minimal T (with quantifier elimination) which is a finite cover of a theory with DMP has generic automorphisms (i.e. the theory T ∪{σ is an automorphism} has a model companion) iff T itself has DMP. They conjectured that the same is true of any strongly minimal theory.