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Stability theory for topological logic, with applications to topological modules1

Published online by Cambridge University Press:  12 March 2014

T. G. Kucera
T. G. Kucera*
Affiliation:
Department of Mathematics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 0W0, Canada
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Extract

In this paper I show how to develop stability theory within the context of the topological logic first introduced by McKee [Mc 76], Garavaglia [G 78] and Ziegler [Z 76]. I then study some specific applications to topological modules; in particular I prove two quantifier élimination theorems, one a generalization of a result of Garavaglia.

In the first section I present a summary of basic results on topological model theory, mostly taken from the book of Flum and Ziegler [FZ 80]. This is done primarily to fix notation, but I also introduce the notion of an Lt-elementary substructure. The important point with this concept, as with many others, appears to be to allow only individuals to appear as parameters, not open sets.

In the second section I begin the study of stability theory for Lt. I first develop a translation of the topological language Lt into an ordinary first-order language L*. The first main theorem is (2.3), which shows that the translation is faithful to the model-theoretic content of Lt, and provides the necessary tools for studying Lt theories in the context of ordinary first-order logic. The translation allows me to consider individual stability theory for Lt: the stability-theoretic study of those types of Lt in which only individual variables occur freely and in which only individuals occur as parameters. I originally developed this stability theory entirely within Lt; the fact that the theorems and their proofs were virtually identical to those in ordinary first order logic suggested the reduction from Lt to L*.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 3 , September 1986 , pp. 755 - 769
Copyright
Copyright © Association for Symbolic Logic 1986

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Footnotes

1

This paper is based on Chapter IV of my doctoral dissertation [Ku 84] prepared under the direction of Dr. M. Makkai at McGill University. I would like to thank Dr. Makkai for his patient and stimulating guidance. I would also like to acknowledge the financial support received through scholarships and grants from the Natural Sciences and Engineering Research Council of Canada (NSERC) and through FCAC Québec.

References

REFERENCES

[Ban 77]Bankston, P., Ultraproducts in topology, General Topology and Its Applications, vol. 7 (1977), pp. 283–308.CrossRefGoogle Scholar
[Ba 76]Baur, W., Elimination of quantifiers for modules, Israel Journal of Mathematics, vol. 25 (1976), pp. 64–70.CrossRefGoogle Scholar
[CK 73]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[CS 81]Cherlin, G. and Schmitt, P. H., Undecidable theories of topological abelian groups, this Journal, vol. 46 (1981), pp. 761–772.Google Scholar
[CS82]Cherlin, G. and Schmitt, P. H., Locally pure topological abelian groups: elementary invariants, preprint, Mathematisches Institut, Universität Heidelberg, Heidelberg, 1982 (published in Annals of Pure and Applied Logic, vol. 24 (1983), pp. 49–85).Google Scholar
[F 77]Fisher, E., Abelian structures. I, Abelian group theory (Arnold, D.et al., editors), Lecture Notes in Mathematics, vol. 616, Springer-Verlag, Berlin, 1977, pp. 270–322.CrossRefGoogle Scholar
[FZ 80]Flum, J. and Ziegler, M., Topological model theory, Lecture Notes in Mathematics, vol. 769, Springer-Verlag, Berlin, 1980.CrossRefGoogle Scholar
[G 78]Garavaglia, S., Model theory of topological structures, Annals of Mathematical Logic, vol. 14 (1978), pp. 13–37.CrossRefGoogle Scholar
[HH 81]Harnik, V. and Harrington, L., Fundamentals of forking, Report No. 11/81, Institute for Advanced Studies, The Hebrew University, Jerusalem (published in Annals of Pure and Applied Logic, vol. 26 (1984), pp. 245–286).CrossRefGoogle Scholar
[Ke 55]Kelley, J. L., General topology, Van Nostrand, Princeton, New Jersey, 1955.Google Scholar
[Ku 84]Kucera, T. G., Applications of stability-theoretical methods to theories of modules, Ph. D. Thesis, McGill University, Montréal, 1984.Google Scholar
[Ku85]Kucera, T. G., Totally transcendental theories of modules: decomposition of models and types, preprint, University of Saskatchewan, Saskatoon, 1985.Google Scholar
[LP 79]Lascar, D. and Poizat, B., Introduction to forking, this Journal, vol. 44 (1979), pp. 330–350.Google Scholar
[M 81]Makkai, M., A survey of basic stability theory, with particular emphasis on orthogonality and regular types, preprint, McGill University, Montréal, 1981 (published in Israel Journal of Mathematics, vol. 49 (1984), pp. 181–238).Google Scholar
[Mc76]McKee, T., Sentences preserved between equivalent topological bases, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 22 (1976), pp. 79–84.CrossRefGoogle Scholar
[Mo 75]Monk, L., Elementary-recursive decision procedures, Ph. D. Thesis, University of California, Berkeley, California, 1975.Google Scholar
[Sa 70]Sabbagh, G., Aspects logiques de la pureté dans les modules, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Series A et B, vol. 271 (1970), pp. A909–A912.Google Scholar
[Z 76]Ziegler, M., A language for topological structures which satisfies a lindström theorem, Bulletin of the American Mathematical Society, vol. 82 (1976), pp. 568–570.CrossRefGoogle Scholar
[Z81]Ziegler, M., Model theory of modules, preprint, 1981 (published in Annals of Pure and Applied Logic, vol. 26 (1984), pp. 149–213).CrossRefGoogle Scholar