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S-homogeneity and automorphism groups

Published online by Cambridge University Press:  12 March 2014

Elisabeth Bouscaren  and
Michael C. Laskowski
Elisabeth Bouscaren
Affiliation:
UFR de Mathematiques, Université de Paris VII, 75251 Paris Cedex 05, France, E-mail: elibou@logique.jussieu.fr
Michael C. Laskowski
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742, E-mail: mcl@math.umd.edu
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Abstract

We consider the question of when, given a subset A of M, the setwise stabilizer of the group of automorphisms induces a closed subgroup on Sym(A). We define s-homogeneity to be the analogue of homogeneity relative to strong embeddings and show that any subset of a countable, s-homogeneous, ω-stable structure induces a closed subgroup and contrast this with a number of negative results. We also show that for ω-stable structures s-homogeneity is preserved under naming countably many constants, but under slightly weaker conditions it can be lost by naming a single point.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 4 , December 1993 , pp. 1302 - 1322
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Baldwin, J., Fundamentals of stability theory, Springer-Verlag, New York, 1988.CrossRefGoogle Scholar
[2]Bouscaren, E., Dimensional order property and pairs of models, Annals of Pure and Applied Logic, vol. 41 (1989), pp. 205–231.CrossRefGoogle Scholar
[3]Bouscaren, E. and Lascar, D., Countable models of nonmultidimensional ℵ0-stable theories, this Journal, vol. 48 (1983), pp. 197–205.Google Scholar
[4]Buechler, S., Locally modular theories of finite rank, Annals of Pure and Applied Logic, vol. 30 (1986), pp. 83–94.CrossRefGoogle Scholar
[5]Buechler, S., The geometry of weakly minimal types, this Journal, vol. 50 (1985), pp. 1044–1053.Google Scholar
[6]Chatzidakis, Z., Model theory of profinite groups having the Iwasawa property, preprint.Google Scholar
[7]Harrington, L. and Makkai, M., An exposition of Shelah's ‘Main Gap’-counting uncountable models of ω-stable and superstable theories, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 139–177.CrossRefGoogle Scholar
[8]Kueker, D. and Steitz, P., Stabilizers of definable sets in homogeneous models, preprint.Google Scholar
[9]Lascar, D., Stability in model theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 36, Longman Scientific & Technical, Harlow; Wiley, New York, 1987.Google Scholar
[10]Lascar, D., Forum Mathematicum, vol. 4 (1992), pp. 243–255.Google Scholar
[11]Makkai, M., A survey of basic stability theory with particular emphasis on orthogonality and regular types, Israel Journal of Mathematics, vol. 49 1–3 (1984), pp. 181–238.CrossRefGoogle Scholar
[12]Pillay, A., Weakly homogeneous models, Proceedings of the American Mathematical Society, vol. 89 (1983), pp. 660–672.Google Scholar
[13]Pillay, A., An introduction to stability theory, Oxford University Press, Oxford, 1983.Google Scholar
[14]Poizat, B., Cours de théorie des modèles, Nur al-Mantiq wal-Ma'rifah, Villeurbanne, France, 1985.Google Scholar
[15]Shelah, S., Classification theory, North-Holland, Amsterdam, 1978.Google Scholar
[16]Shelah, S., Harrington, L. and Makkai, M., A proof of the Vaught conjecture for ω-stable theories, Proceedings of the 1980/1 Jerusalem Model Theory Year, Israel Journal of Mathematics, vol. 49 (1984), pp. 259–280.Google Scholar