Simple generic structures

AbstractA study of smooth classes whose generic structures have simple theory is carried out in a spirit similar to Hrushovski (Ann. Pure Appl. Logic 62 (1993) 147; Simplicity and the Lascar group, preprint, 1997) and Baldwin–Shi (Ann. Pure Appl. Logic 79 (1) (1996) 1). We attach to a smooth class 〈K0,≺〉 of finite L-structures a canonical inductive theory TNat, in an extension-by-definition of the language L. Here TNat and the class of existentially closed models of (TNat)∀=T+,EX(T+), play an important role in description of the theory of the 〈K0,≺〉-generic. We show that if M is the 〈K0,≺〉-generic then M∈EX(T+). Furthermore, if this class is an elementary class then Th(M)=Th(EX(T+)). The investigations by Hrushovski (preprint, 1997) and Pillay (Forking in the category of existentially closed structures, preprint, 1999), provide a general theory for forking and simplicity for the nonelementary classes, and using these ideas, we show that if 〈K0,≺〉, where ≺∈{⩽,⩽∗}, has the joint embedding property and is closed under the Independence Theorem Diagram then EX(T+) is simple. Moreover, we study cases where EX(T+) is an elementary class. We introduce the notion of semigenericity and show that if a 〈K0,≺〉-semigeneric structure exists then EX(T+) is an elementary class and therefore the L-theory of 〈K0,≺〉-generic is near model complete. By this result we are able to give a new proof for a theorem of Baldwin and Shelah (Trans. AMS 349 (4) (1997) 1359). We conclude this paper by giving an example of a generic structure whose (full) first-order theory is simple