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The classification of small types of rank ω, Part I

Published online by Cambridge University Press:  12 March 2014

Steven Buechler  and
Colleen Hoover
Steven Buechler
Affiliation:
Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame, IN 46556-4618, USA, E-Mail: buechler.l@nd.edu
Colleen Hoover
Affiliation:
Department of Mathematics, St. Mary's college, Notre Dame. IN 46556, USA, E-Mail: choover@saintmarys.edu
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Abstract.

Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught's conjecture. As an application of the methods, we prove

Theorem A. Let G be a superstate group of U-rank ω such that the generics of G are locally modular and Th(G) has few countable models. Let G− be the group of nongeneric elements of G. G+ = Go + G−. Let Π = {q ∈ S(∅): U(q) < ω}. For any countable model M of Th(G) there is a finite A ⊂ M such thai M is almost atomic over A ∪ (G+ ∩ M) ∪ ⋃p∈Πp(M).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 4 , December 2001 , pp. 1884 - 1898
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

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