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Paper IPM / M / 728  


Abstract:  
A study of smooth classes whose generic structures have simple theory is carried out in a similar spirit to Hrushovski [5], [6] and BaldwinShi [3]. We attach to a smooth class 〈K_{0}, \prec〉 of finite
Lstructures a canonical inductive theory T_{Nat}, in an extensionbydefinition of the language L. Here T_{Nat} and its class of existentially closed models, EX(T_{Nat}), play an important role in description of the
theory of the 〈K_{0}, \prec〉generic. We show that if M is the 〈K_{0}, \prec〉generic then M ∈ EX(T_{Nat}). Furthermore, if this class is an elementary class then Th(M)=Th(EX(T_{Nat})). 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(T_{Nat}) and provide a
sufficient condition, namely the Independence Theorem Diagram, for which this class is simple. We show that if 〈K_{0}, \prec〉, where \prec ∈ { ≤ , ≤ ^{*}}, has the joint embedding property and is closed under the Independence Theorem Diagram then EX(T_{Nat}) is simple. Moreover, we study cases
where EX(T_{Nat}) is an elementary class. We introduce the notion of semigenericity and show that if a 〈K_{0}, \prec〉semigeneric structure exists then EX(T_{Nat}) is an elementary class and therefore the Ltheory of
〈K_{0}, \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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