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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Simple homogeneous models

Authors: Steven Buechler and Olivier Lessmann
Journal: J. Amer. Math. Soc. 16 (2003), 91-121
MSC (2000): Primary 03C45
Published electronically: October 8, 2002
MathSciNet review: 1937201
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Abstract: Geometrical stability theory is a powerful set of model-theoretic tools that can lead to structural results on models of a simple first-order theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theory (a saturated model) in which the Stone space topology on ultrafilters of definable relations is compact. Here we operate in the more general setting of homogeneous models, which typically have noncompact Stone topologies. A structure $M$ equipped with a class of finitary relations $\mathcal {R}$ is strongly $\lambda$-homogeneous if orbits under automorphisms of $(M,\mathcal {R})$ have finite character in the following sense: Given $\alpha$ an ordinal $<\lambda \leq |M|$ and sequences $\bar {a}=\{ a_i:\:i<\alpha \}$, $\bar {b}=\{ b_i:\:i<\alpha \}$ from $M$, if $(a_{i_1},\dots ,a_{i_n})$ and $(b_{i_1},\dots ,b_{i_n})$ have the same orbit, for all $n$ and $i_1<\dots <i_n<\alpha$, then $f(\bar {a})=\bar {b}$ for some automorphism $f$ of $(M,\mathcal {R})$. In this paper strongly $\lambda$-homogeneous models $(M,\mathcal {R})$ in which the elements of $\mathcal {R}$ induce a symmetric and transitive notion of independence with bounded character are studied. This notion of independence, defined using a combinatorial condition called “dividing”, agrees with forking independence when $(M,\mathcal {R})$ is saturated. A concept central to the development of stability theory for saturated structures, namely parallelism, is also shown to be well-behaved in this setting. These results broaden the scope of the methods of geometrical stability theory.

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Additional Information

Steven Buechler
Affiliation: Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame, Indiana 46556

Olivier Lessmann
Affiliation: Mathematical Institute, 24-29 St. Giles, Oxford University, Oxford OX1 3LB, United Kingdom

Keywords: Stability theory, simple theories, nonelementary classes
Received by editor(s): September 22, 2001
Received by editor(s) in revised form: August 19, 2002
Published electronically: October 8, 2002
Additional Notes: Research of the first author was partially supported by the NSF
Article copyright: © Copyright 2002 American Mathematical Society