Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-29T14:29:16.840Z Has data issue: false hasContentIssue false

S-homogeneity and automorphism groups

Published online by Cambridge University Press:  12 March 2014

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

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
Copyright
Copyright © Association for Symbolic Logic 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 205231.CrossRefGoogle Scholar
[3]Bouscaren, E. and Lascar, D., Countable models of nonmultidimensional ℵ0-stable theories, this Journal, vol. 48 (1983), pp. 197205.Google Scholar
[4]Buechler, S., Locally modular theories of finite rank, Annals of Pure and Applied Logic, vol. 30 (1986), pp. 8394.CrossRefGoogle Scholar
[5]Buechler, S., The geometry of weakly minimal types, this Journal, vol. 50 (1985), pp. 10441053.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. 139177.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. 243255.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. 181238.CrossRefGoogle Scholar
[12]Pillay, A., Weakly homogeneous models, Proceedings of the American Mathematical Society, vol. 89 (1983), pp. 660672.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. 259280.Google Scholar