## Simple homogeneous models

HTML articles powered by AMS MathViewer

- by Steven Buechler and Olivier Lessmann;
- J. Amer. Math. Soc.
**16**(2003), 91-121 - DOI: https://doi.org/10.1090/S0894-0347-02-00407-1
- Published electronically: October 8, 2002
- PDF | Request permission

## 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.

## References

- A. Berenstein and S. Buechler. Simple stable homogeneous expansions of Hilbert spaces. preprint, 2002.
- Alexander Berenstein.
*Dependence relations on homogeneous groups and homogeneous expansions of Hilbert spaces*. PhD thesis, University of Notre Dame, 2002. - Steven Buechler, Anand Pillay, and Frank Wagner,
*Supersimple theories*, J. Amer. Math. Soc.**14**(2001), no. 1, 109–124. MR**1800350**, DOI 10.1090/S0894-0347-00-00350-7 - Steven Buechler,
*Essential stability theory*, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1996. MR**1416106**, DOI 10.1007/978-3-642-80177-8 - Itay Ben-Yaacov. Discouraging results for ultraimaginary independence theory. preprint, 2002.
- Zoé Chatzidakis and Ehud Hrushovski,
*Model theory of difference fields*, Trans. Amer. Math. Soc.**351**(1999), no. 8, 2997–3071. MR**1652269**, DOI 10.1090/S0002-9947-99-02498-8 - G. Cherlin, L. Harrington, and A. H. Lachlan,
*$\aleph _0$-categorical, $\aleph _0$-stable structures*, Ann. Pure Appl. Logic**28**(1985), no. 2, 103–135. MR**779159**, DOI 10.1016/0168-0072(85)90023-5 - C. C. Chang and H. J. Keisler,
*Model theory*, Studies in Logic and the Foundations of Mathematics, Vol. 73, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR**409165** - Rami Grossberg and Olivier Lessmann. Shelah’s stability spectrum and homogeneity spectrum in finite diagrams.
*Archive for Mathematical Logic*. to appear. - Bradd Hart, Byunghan Kim, and Anand Pillay,
*Coordinatisation and canonical bases in simple theories*, J. Symbolic Logic**65**(2000), no. 1, 293–309. MR**1782121**, DOI 10.2307/2586538 - Tapani Hyttinen and Olivier Lessmann. A rank for the class of elementary submodels of a superstable homogeneous model. Journal of Symbolic Logic, to appear.
- Ehud Hrushovski,
*Stability and its uses*, Current developments in mathematics, 1996 (Cambridge, MA), Int. Press, Boston, MA, 1997, pp. 61–103. MR**1724945** - Ehud Hrushovski,
*Geometric model theory*, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 281–302. MR**1648035** - Ehud Hrushovski. Simplicity and the lascar group. preprint, 1998.
- Edwin Hewitt and Karl Stromberg,
*Real and abstract analysis. A modern treatment of the theory of functions of a real variable*, Springer-Verlag, New York, 1965. MR**188387** - S. Losinsky,
*Sur le procédé d’interpolation de Fejér*, C. R. (Doklady) Acad. Sci. URSS (N.S.)**24**(1939), 318–321 (French). MR**2001** - Ehud Hrushovski and Boris Zilber,
*Zariski geometries*, J. Amer. Math. Soc.**9**(1996), no. 1, 1–56. MR**1311822**, DOI 10.1090/S0894-0347-96-00180-4 - José Iovino,
*Stable Banach spaces and Banach space structures. I. Fundamentals*, Models, algebras, and proofs (Bogotá, 1995) Lecture Notes in Pure and Appl. Math., vol. 203, Dekker, New York, 1999, pp. 77–95. MR**1686915** - H. Jerome Keisler,
*Model theory for infinitary logic. Logic with countable conjunctions and finite quantifiers*, Studies in Logic and the Foundations of Mathematics, Vol. 62, North-Holland Publishing Co., Amsterdam-London, 1971. MR**344115** - Byunghan Kim,
*Forking in simple unstable theories*, J. London Math. Soc. (2)**57**(1998), no. 2, 257–267. MR**1644264**, DOI 10.1112/S0024610798005985 - B. Kim and A. Pillay. Simple theories.
*Annals of Pure and Applied Logic*. to appear. - Daniel Lascar,
*On the category of models of a complete theory*, J. Symbolic Logic**47**(1982), no. 2, 249–266. MR**654786**, DOI 10.2307/2273140 - David Marker and Anand Pillay,
*Differential Galois theory. III. Some inverse problems*, Illinois J. Math.**41**(1997), no. 3, 453–461. MR**1458184** - Anand Pillay,
*Differential Galois theory. II*, Ann. Pure Appl. Logic**88**(1997), no. 2-3, 181–191. Joint AILA-KGS Model Theory Meeting (Florence, 1995). MR**1600903**, DOI 10.1016/S0168-0072(97)00021-3 - Anand Pillay,
*Differential Galois theory. I*, Illinois J. Math.**42**(1998), no. 4, 678–699. MR**1649893** - Anand Pillay. Forking in the category of existentially closed structures. preprint, 1998.
- Saharon Shelah,
*Finite diagrams stable in power*, Ann. Math. Logic**2**(1970/71), no. 1, 69–118. MR**285374**, DOI 10.1016/0003-4843(70)90007-0 - Saharon Shelah,
*The lazy model-theoretician’s guide to stability*, Logique et Anal. (N.S.)**18**(1975), no. 71-72, 241–308. MR**539969** - Saharon Shelah,
*Simple unstable theories*, Ann. Math. Logic**19**(1980), no. 3, 177–203. MR**595012**, DOI 10.1016/0003-4843(80)90009-1 - S. Shelah,
*Classification theory and the number of nonisomorphic models*, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR**1083551** - Frank Wagner,
*Hyperdefinable groups in simple theories*, J. Math. Log.**1**(2001), no. 1, 125–172. MR**1838357**, DOI 10.1142/S0219061301000053 - Boris Zilber,
*Uncountably categorical theories*, Translations of Mathematical Monographs, vol. 117, American Mathematical Society, Providence, RI, 1993. Translated from the Russian by D. Louvish. MR**1206477**, DOI 10.1090/mmono/117

## Bibliographic 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
- 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
- © Copyright 2002 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**16**(2003), 91-121 - MSC (2000): Primary 03C45
- DOI: https://doi.org/10.1090/S0894-0347-02-00407-1
- MathSciNet review: 1937201