Hostname: page-component-5d59c44645-klj7v Total loading time: 0 Render date: 2024-02-22T00:32:36.562Z Has data issue: false hasContentIssue false

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

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

[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