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Generic stability and stability

By Hans Adler and Enrique Casanovas


We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from [3] on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of p (m) for all m. We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If p(x) ∈ S(B) is stable and does not fork over A then p ↾ A is stable. (They had solved some special cases.) 1 Stable and generically stable types Our notation is standard. We work with an arbitrary complete theory T in language L. C denotes a ‘monster model’. M denotes a small elementary submodel, and A, B,... denote small subsets. L(C) denotes the collection of formulas with parameters from C, likewise L(A) etc. We sometimes say A-invariant for Aut(C/A)-invariant. By a global type we mean a complete type over C. We assume familiarity with notions from model theory such as heir, coheir, definable type, forking. The book [7] is a good reference, but see also [5]. Stable types Definition 1.1 Let π(x) be a partial type over a set A of parameters. π is stable if all complete extensions p(x) ∈ S(B) of π over any set B ⊇ A are definable. The definition of stable partial type goes back to Lascar and Poizat. Many other equivalent formulations of this notion are known, which we now mention. These should be considered well known, but Section 10 of [1] contains proofs and/or references for the next three Remarks/Facts. Remark 1.2 The following are equivalent for any partial type π(x) over A: 1. π(x) is stable. 2. For every B ⊇ A there are at most |B | |T | types p(x) ∈ S(B) extending π(x)

Year: 2012
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