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Paper   IPM / M / 728
School of Mathematics
  Title:   Simple generic structures
  Author(s):  M. Pourmahdian
  Status:   Published
  Journal: Ann. Pure Appl. Logic
  Vol.:  121
  Year:  2003
  Pages:   227-260
  Supported by:  IPM
A study of smooth classes whose generic structures have simple theory is carried out in a similar spirit to Hrushovski [5], [6] and Baldwin-Shi [3]. We attach to a smooth class 〈K0, \prec〉 of finite L-structures a canonical inductive theory TNat, in an extension-by-definition of the language L. Here TNat and its class of existentially closed models, EX(TNat), play an important role in description of the theory of the 〈K0, \prec〉-generic. We show that if M is the 〈K0, \prec〉-generic then MEX(TNat). Furthermore, if this class is an elementary class then Th(M)=Th(EX(TNat)). The investigations by Hrshovski [6] and Pillay [14], provide a general theory for forking and simplicity for the nonelementary classes and using these ideas, we study simplicity of EX(TNat) and provide a sufficient condition, namely the Independence Theorem Diagram, for which this class is simple. We show that if 〈K0, \prec〉, where \prec ∈ { ≤ , ≤ *}, has the joint embedding property and is closed under the Independence Theorem Diagram then EX(TNat) is simple. Moreover, we study cases where EX(TNat) is an elementary class. We introduce the notion of semigenericity and show that if a 〈K0, \prec〉-semigeneric structure exists then EX(TNat) is an elementary class and therefore the L-theory of 〈K0, \prec〉-generic is near model complete. By this result we are able to give a new proof for a theorem of Baldwin and Shelah [1]. We conclude this paper by giving an example of a generic structure whose (full) first order theory is simple.

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