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Subsets of superstable structures are weakly benign

Published online by Cambridge University Press:  12 March 2014

Bektur Baizhanov
Affiliation:
Institute for Problems of Informatics and Control, Pushkin STR. 125, Almaty 480100, Kazakhstan, E-mail: baizhanov@ipic.kz
John T. Baldwin
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street Chicago, IL 60607., USA, E-mail: jbaldwin@math.uic.edu
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel Rutgers University, Mathematics Department, New Brunswick, NJ, USA

Extract

Baizhanov and Baldwin [1] introduce the notions of benign and weakly benign sets to investigate the preservation of stability by naming arbitrary subsets of a stable structure. They connect the notion with work of Baldwin, Benedikt, Bouscaren, Casanovas, Poizat, and Ziegler. Stimulated by [1], we investigate here the existence of benign or weakly benign sets.

Definition 0.1. (1) The set A is benign in M if for every α, β ∊ M if p = tp(α/A) = tp(β/A) then tp*(α/A) = tp*(β/A) where the *-type is the type in the language L* with a new predicate P denoting A.

(2) The set A is weakly benign in M if for every α,β ∊ M if p = stp(α/A) = stp(β/A) then tp*(α/A) = tp*(β/A) where the *-type is the type in language with a new predicate P denoting A.

Conjecture 0.2 (too optimistic). If M is a model of stable theory T and A ⊆ M then A is benign.

Shelah observed, after learning of the Baizhanov-Baldwin reductions of the problem to equivalence relations, the following counterexample.

Lemma 0.3. There is an ω-stable rank 2 theory T with ndop which has a model M and set A such that A is not benign in M.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2005

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References

[1]Baizhanov, B. and Baldwin, J. T., Local homogeneity, this Journal, vol. 69 (2004), pp. 12431260.Google Scholar
[2]Baldwin, J. T., Fundamentals of stability theory, Springer-Verlag, 1988.CrossRefGoogle Scholar
[3]Bouscaren, E., Dimensional order property and pairs of models, Annals of Pure and Applied Logic, vol. 41 (1989), pp. 205231.CrossRefGoogle Scholar
[4]Shelah, S., Classification theory and the number of nonisomorphic models, second ed., North-Holland, 1991.Google Scholar
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