Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-23T23:04:40.741Z Has data issue: false hasContentIssue false
  • English
  • Français

Expansion of a model of a weakly o-minimal theory by a family of unary predicates

Published online by Cambridge University Press:  12 March 2014

Bektur Sembiuly Baizhanov
Bektur Sembiuly Baizhanov*
Affiliation:
Institute of Informatics, and Control Problems, Ul.Poushkina, 125, 480100 Almaty, Kazakhstan, E-mail: baizhanov@hotmail.com, E-mail: baijan@ipic.academ.alma-ata.su
Get access Rights & Permissions [Opens in a new window]

Abstract

A subset A ⊆ M of a totally ordered structure M is said to be convex, if for any a, b ∈ A: [a < b → ∀t (a < tb → t ∈ A)]. A complete theory of first order is weakly o-minimal (M. Dickmann [D]) if any model M is totally ordered by some ∅-definable formula and any subset of M which is definable with parameters from M is a finite union of convex sets. We prove here that for any model M of a weakly o-minimal theory T. any expansion M+ of M by a family of unary predicates has a weakly o-minimal theory iff the set of all realizations of each predicate is a union of a finite number of convex sets (Theorem 63). that solves the Problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1382 - 1414
Copyright
Copyright © Association for Symbolic Logic 2001

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]Baldwin, J. T., Fundamentals of stability theory, Springer-Verlag, New York, 1988.CrossRefGoogle Scholar
[BB1]Baizhanov, B. S., Expansion of an o-minimal model by unary convex predicates, Researches in theory of algebraic systems (Nurmagambetov, T., editor), Karaganda State University, 1995, pp. 3–23.Google Scholar
[BB2]Baizhanov, B. S., Classification of one-types in weakly o-minimal theories and its corollaries, preprint, 1996.Google Scholar
[BB3]Baizhanov, B. S., Orthogonality of one-types in weakly o-minimal theories, Algebra and model theory II (Pinus, A. G. and Ponomaryov, K. N., editors), Novosibirsk State Technical University, 1999, pp. 3–28.Google Scholar
[CK]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[D]Dickmann, M., Elimination of quantifiers for ordered valution rings, Proceedings 3rd Easter model theory conference at Gross Koris (Berlin), 1985.Google Scholar
[EBBP]Baisalov, E. and Poizat, B., Paires de structure o-minimal, prepublications de l'Institut Girard Desargues, UPRES–A 5028, 1996, N18, pp. 1–9.Google Scholar
[LD]van den Dries, L., Remarks on Tarski's problem concerning (R, +, *, exp), manuscript, 1983.CrossRefGoogle Scholar
[LM]Mayer, L., Vaught's conjecture for o-minimal theories, this Journal, vol. 53 (1988), pp. 146–159.Google Scholar
[LS]Laskovski, M. and Steinhorn, Ch., On o-minimal expansions of archimedian ordered group, pp. 1–21, preprint, 1994.Google Scholar
[MMS]Macpherson, D., Marker, D., and Steinhorn, Ch., Weakly o-minimal structures and real closed fields, preprint, 1993.Google Scholar
[M]Marker, D., Omitting types in o-minimal theories, this Journal, vol. 51 (1986), pp. 63–74.Google Scholar
[MS]Marker, D. and Steinhorn, Ch., Definable types in o-minimal theories, this Journal, vol. 59 (1994), pp. 185–198.Google Scholar
[P]Pillay, A., Definability of types, and pairs of o-minimal structures, this Journal, vol. 59 (1994), pp. 1400–1409.Google Scholar
[PS]Pillay, A. and Steinhorn, Ch., Definable sets in ordered structures. 1, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565–592.CrossRefGoogle Scholar
[S]Shelah, S., Classification Theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar