Simple generic structures

Annals of Pure and Applied Logic 121 (2-3):227-260 (2003)
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Abstract

A study of smooth classes whose generic structures have simple theory is carried out in a spirit similar to Hrushovski 147; Simplicity and the Lascar group, preprint, 1997) and Baldwin–Shi 1). We attach to a smooth class K0, of finite -structures a canonical inductive theory TNat, in an extension-by-definition of the language . Here TNat and the class of existentially closed models of =T+,EX, play an important role in description of the theory of the K0,-generic. We show that if M is the K0,-generic then MEX. Furthermore, if this class is an elementary class then Th=Th). The investigations by Hrushovski and Pillay , 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 is simple. Moreover, we study cases where EX is an elementary class. We introduce the notion of semigenericity and show that if a K0,-semigeneric structure exists then EX is an elementary class and therefore the -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 1359). We conclude this paper by giving an example of a generic structure whose first-order theory is simple

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Citations of this work

Some Remarks on Generic Structures.David M. Evans & Mark Wing Ho Wong - 2009 - Journal of Symbolic Logic 74 (4):1143-1154.
Pseudofiniteness in Hrushovski Constructions.Ali N. Valizadeh & Massoud Pourmahdian - 2020 - Notre Dame Journal of Formal Logic 61 (1):1-10.
Consistent amalgamation for þ-forking.Clifton Ealy & Alf Onshuus - 2014 - Annals of Pure and Applied Logic 165 (2):503-519.

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References found in this work

Simple theories.Byunghan Kim & Anand Pillay - 1997 - Annals of Pure and Applied Logic 88 (2-3):149-164.
A new strongly minimal set.Ehud Hrushovski - 1993 - Annals of Pure and Applied Logic 62 (2):147-166.
Stable generic structures.John T. Baldwin & Niandong Shi - 1996 - Annals of Pure and Applied Logic 79 (1):1-35.
On generic structures.D. W. Kueker & M. C. Laskowski - 1992 - Notre Dame Journal of Formal Logic 33 (2):175-183.
From stability to simplicity.Byunghan Kim & Anand Pillay - 1998 - Bulletin of Symbolic Logic 4 (1):17-36.

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