Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T22:23:11.623Z Has data issue: false hasContentIssue false
  • English
  • Français

Abelian groups with modular generic

Published online by Cambridge University Press:  12 March 2014

James Loveys
James Loveys*
Affiliation:
Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 2K6, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

Let G be a stable abelian group with regular modular generic. We show that either

1. there is a definable nongeneric K ≤ G such that G/K has definable connected component and so strongly regular generics, or

2. distinct elements of the division ring yielding the dependence relation are represented by subgroups of G × G realizing distinct strong types (when regarded as elements of Geq).

In the latter case one can choose almost 0-definable subgroups representing the elements of the division ring. We find a bound ((G: G0)) for the size of the division ring in case G has no definable subgroup K so that G/K is infinite with definable connected component. We show in case (2) that the group G/H, where H consists of all nongeneric points of G, inherits a weakly minimal group structure from G naturally, and Th(G/H) is independent of the particular model G as long as G/H is infinite.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 1 , March 1991 , pp. 250 - 259
Copyright
Copyright © Association for Symbolic Logic 1991

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

[B]Buechler, S., The geometry of weakly minimal types, this Journal, vol. 50 (1985), pp. 1044–1053.Google Scholar
[H1]Hrushovski, E., Locally modular regular types, Classification theory (proceedings of the U.S.-Israel workshop on model theory in mathematical logic, Chicago, 1985; Baldwin, J., editor), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin, 1987, pp. 132–164.Google Scholar
[H2]Hrushovski, E., Unidimensional theories are superstable, Annals of Pure and Applied Logic (submitted).Google Scholar
[H3]Hrushovski, E., Almost orthogonal regular types, Annals of Pure and Applied Logic (submitted).Google Scholar
[HL]Hrushovski, E. and Loveys, J., Weakly minimal groups of bounded exponent (in preparation).Google Scholar
[HS]Hrushovski, E. and Shelah, S., A dichotomy theorem for regular types (to appear).Google Scholar
[L1]Loveys, J., Weakly minimal groups of unbounded exponent, this Journal, vol. 55 (1990), pp. 928–937.Google Scholar
[L2]Loveys, J., Locally finite weakly minimal theories, Annals of Pure and Applied Logic (to appear).Google Scholar
[M]Makkai, M., A survey of basic stability theory, with particular emphasis on orthogonality and regular types, Israel Journal of Mathematics, vol. 49 (1985), pp. 181–238.CrossRefGoogle Scholar
[Pi]Pillay, A., A note on subgroups of the automorphism group of a saturated model, and regular types, this Journal, vol. 54 (1989), pp. 858–864.Google Scholar
[P1]Poizat, B., Groupes stables, avec types génériques régulières, this Journal, vol. 48 (1983), pp. 339–355.Google Scholar
[P2]Poizat, B., Groupes stables, Nur al-Mantiq wal-Ma'rifah, Villeurbanne, 1987.Google Scholar
[Sh]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar