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On dividing chains in simple theories. (English) Zbl 1076.03022
Summary: Dividing chains have been used as conditions to isolate adequate subclasses of simple theories. In the first part of this paper we present an introduction to the area. We give an overview on fundamental notions and present proofs of some of the basic and well-known facts related to dividing chains in simple theories. In the second part we discuss various characterizations of the subclass of low theories. Our main theorem generalizes and slightly extends a well-known fact about the connection between dividing chains and Morley sequences (in our case: independent sequences). Moreover, we are able to give a proof that is shorter than the original one. This result motivates us to introduce a special property of formulas concerning independent dividing chains: For any dividing chain there exists an independent dividing chain of the same length. We study this property in the context of low, short and $$\omega$$-categorical simple theories, outline some examples and define subclasses of low and short theories, which imply this property. The results give rise to further studies of the relationships between some subclasses of simple theories.
MSC:
 03C45 Classification theory, stability, and related concepts in model theory
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References:
 [1] Ben-Yaacov, I., Pillay, A., Vassiliev, E.: Lovely pairs of models. Annals of Pure and Applied Logic 122, 235–261 (2003) · Zbl 1030.03026 [2] Buechler, St.: Lascar strong types in some simple theories. The Journal of Symbolic Logic 64, 817–824 (1999) · Zbl 0930.03035 [3] Buechler, St., Pillay, A., Wagner, F.O.: Supersimple theories. Journal of the American Mathematical Society 14, 109–124 (2001) · Zbl 0963.03057 [4] Casanovas, E.: The number of types in simple theories. Annals of Pure and Applied Logic 98, 69–86 (1999) · Zbl 0939.03039 [5] Casanovas, E., Kim, B.: A supersimple nonlow theory. Notre Dame Journal of Formal Logic 39, 507–518 (1998) · Zbl 0973.03048 [6] Casanovas, E., Wagner, F.O.: Local supersimplicity and related concepts. The Journal of Symbolic Logic 67, 744–758 (2002) · Zbl 1013.03035 [7] Kim, B.: Simple First Order Theories. Ph.D. thesis, University of Notre Dame. USA, 1996 [8] Wagner, F.O.: Simple Theories. Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000 · Zbl 0948.03032
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