Simple homogeneous models

Authors:
Steven Buechler and Olivier Lessmann

Journal:
J. Amer. Math. Soc. **16** (2003), 91-121

MSC (2000):
Primary 03C45

DOI:
https://doi.org/10.1090/S0894-0347-02-00407-1

Published electronically:
October 8, 2002

MathSciNet review:
1937201

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Abstract | References | Similar Articles | Additional Information

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 equipped with a class of finitary relations is *strongly -homogeneous* if orbits under automorphisms of have finite character in the following sense: Given an ordinal and sequences , from , if and have the same orbit, for all and , then for some automorphism of . In this paper strongly -homogeneous models in which the elements of 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 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.

**[BB02]**A. Berenstein and S. Buechler.

Simple stable homogeneous expansions of Hilbert spaces.

preprint, 2002.**[Ber02]**Alexander Berenstein.*Dependence relations on homogeneous groups and homogeneous expansions of Hilbert spaces*.

PhD thesis, University of Notre Dame, 2002.**[BPW01]**S. Buechler, A. Pillay, and F. Wagner.

Supersimple theories.*J. Amer. Math. Soc.*, 14:109-124, 2001. MR**2001j:03065****[Bue96]**Steven Buechler.*Essential Stability Theory*.

Perspectives in Mathematical Logic. Springer-Verlag, Berlin/Heidelberg/New York, 1996. MR**98j:03050****[BY02]**Itay Ben-Yaacov.

Discouraging results for ultraimaginary independence theory.

preprint, 2002.**[CH99]**Zoé Chatzidakis and Ehud Hrushovski.

Model theory of difference fields.*Trans. Amer. Math. Soc.*, 351(8):2997-3071, 1999. MR**2000f:03109****[CHL85]**G. Cherlin, L. Harrington, and A. H. Lachlan.

-categorical, -stable structures.*Ann. Pure Appl. Logic*, 28(2):103-135, 1985. MR**86g:03054****[CK73]**C. C. Chang and H. J. Keisler.*Model Theory*.

North Holland/Elsevier, Amsterdam/London/New York, 1973. MR**53:12927****[GL]**Rami Grossberg and Olivier Lessmann.

Shelah's stability spectrum and homogeneity spectrum in finite diagrams.*Archive for Mathematical Logic*.

to appear.**[HKP00]**B. Hart, B. Kim, and A. Pillay.

Coordinatisation and canonical bases in simple theories.*J. of Symbolic Logic*, 65:293-309, 2000. MR**2001j:03066****[HL]**Tapani Hyttinen and Olivier Lessmann.

A rank for the class of elementary submodels of a superstable homogeneous model.

Journal of Symbolic Logic, to appear.**[Hru97]**Ehud Hrushovski.

Stability and its uses.

In*Current developments in mathematics, 1996 (Cambridge, MA)*, pages 61-103. Int. Press, Boston, MA, 1997. MR**2000h:11124****[Hru98]**Ehud Hrushovski.

Geometric model theory.

In*Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998)*, Doc. Math. 1998, Extra Vol. I, pages 281-302 (electronic), 1998. MR**2000b:03120****[Hru]**Ehud Hrushovski.

Simplicity and the lascar group.

preprint, 1998.**[HS65]**Edwin Hewitt and Karl Stromberg.*Real and Abstract Analysis*.

Springer-Verlag, Berlin/ Heidelberg/ New York, 1965. MR**32:5826****[HS00]**Tapani Hyttinen and Saharon Shelah.

Strong splitting in stable homogeneous models.*Ann. Pure Appl. Logic*, 103(1-3):201-228, 2000. MR**2001:03068****[HZ96]**Ehud Hrushovski and Boris Zilber.

Zariski geometries.*J. Amer. Math. Soc.*, 9(1):1-56, 1996. MR**96c:03077****[Iov99]**José Iovino.

Stable Banach spaces and banach space structures i: Fundamentals.

In*Models, Algebra, and Proofs. (Bogota 1995)*, volume 203 of*Lecture notes in pure and applied mathematics*, pages 77-95. Marcel-Dekker, New York, 1999. MR**2000h:03064a****[Kei71]**H. J. Keisler.*Model theory for infinitary logic*.

North-Holland, Amsterdam/New York, 1971. MR**49:8855****[Kim98]**Byunghan Kim.

Forking in simple unstable theories.*Journal of the London Math. Society*, 57(2):257-267, 1998. MR**2000a:03052****[KP]**B. Kim and A. Pillay.

Simple theories.*Annals of Pure and Applied Logic*.

to appear.**[Las82]**Daniel Lascar.

On the category of models of a complete theory.*J. of Symbolic Logic*, 47:249-266, 1982. MR**84g:03055****[MP97]**David Marker and Anand Pillay.

Differential Galois theory. III. Some inverse problems.*Illinois J. Math.*, 41(3):453-461, 1997. MR**99m:12011****[Pil97]**Anand Pillay.

Differential Galois theory. II.*Ann. Pure Appl. Logic*, 88(2-3):181-191, 1997.

Joint AILA-KGS Model Theory Meeting (Florence, 1995). MR**99m:12010****[Pil98]**Anand Pillay.

Differential Galois theory. I.*Illinois J. Math.*, 42(4):678-699, 1998. MR**99m:12009****[Pil]**Anand Pillay.

Forking in the category of existentially closed structures.

preprint, 1998.**[She70]**Saharon Shelah.

Finite diagrams stable in power.*Annals of Pure and Applied Logic*, 2:69-118, 1970. MR**44:2593****[She75]**Saharon Shelah.

The lazy model theorist's guide to stability.

In P. Henrand, editor,*Proc. of a Symp. in Louvain, March, 1975*, volume 71 of*Logique et Analyse*, pages 241-308, 1975. MR**58:27447****[She80]**Saharon Shelah.

Simple unstable theories.*Annals of Pure and Applied Logic*, 19:177-203, 1980. MR**82g:03055****[She90]**Saharon Shelah.*Classification Theory and the number of nonisomorphic models*.

North-Holland, Amsterdam and New York, 1990.

Revised Edition. MR**91k:03085****[Wag01]**Frank Wagner.

Hyperdefinable groups in simple theories.*J. Math. Log.*, 1(1):125-172, 2001. MR**2002f:03068****[Zil93]**Boris Zilber.*Uncountably categorical theories*.

American Mathematical Society, Providence, RI, 1993.

Translated from the Russian by D. Louvish. MR**94h:03059**

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Additional Information

**Steven Buechler**

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

Email:
buechler.1@nd.edu

**Olivier Lessmann**

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

Email:
lessmann@maths.ox.ac.uk

DOI:
https://doi.org/10.1090/S0894-0347-02-00407-1

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