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 Algebra Logika, 2003, Volume 42, Number 2, Pages 194–210 (Mi al25)

$E^*$-Stable Theories

E. A. Palyutin

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: S. Shelah proved that stability of a theory is equivalent to definability of every complete type of that theory. T. Mustafin introduced the concept of being $T^*$-stable, generalizing the notion of being stable. However, $T^*$-stability does not necessitate definability of types. The key result of the present article is proving the definability of types for $E^*$-stable theories. This concept differs from that of being $T^*$-stable by adding the condition of being continuous. As a consequence we arrive at the definability of types over any $P$-sets in $P$-stable theories, which previously was established by T. Nurmagambetov and B. Poizat for types over $P$-models.

Keywords: $E^*$-stable theory, definability of types.

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English version:
Algebra and Logic, 2003, 42:2, 112–120

Bibliographic databases:

UDC: 510.67:512.57

Citation: E. A. Palyutin, “$E^*$-Stable Theories”, Algebra Logika, 42:2 (2003), 194–210; Algebra and Logic, 42:2 (2003), 112–120

Citation in format AMSBIB
\Bibitem{Pal03} \by E.~A.~Palyutin \paper $E^*$-Stable Theories \jour Algebra Logika \yr 2003 \vol 42 \issue 2 \pages 194--210 \mathnet{http://mi.mathnet.ru/al25} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2003629} \zmath{https://zbmath.org/?q=an:1029.03019} \transl \jour Algebra and Logic \yr 2003 \vol 42 \issue 2 \pages 112--120 \crossref{https://doi.org/10.1023/A:1023302423817} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-27544484305} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. E. A. Palyutin, “Elementary Pairs of Primitive Normal Theories”, Algebra and Logic, 43:3 (2004), 179–189
2. E. A. Palyutin, “Stably Definable Classes of Theories”, Algebra and Logic, 44:5 (2005), 326–335
3. M. A. Rusaleev, “Characterization of $(p,1)$-stable theories”, Algebra and Logic, 46:3 (2007), 188–194
4. A. R. Yeshkeyev, “On Jonsson stability and some of its generalizations”, J. Math. Sci., 166:5 (2010), 646–654
5. M. A. Rusaleev, “Generalized stability of torsion-free Abelian groups”, Algebra and Logic, 50:2 (2011), 161–170
6. B. S. Baizhanov, V. V. Verbovskii, “$o$-stable theories”, Algebra and Logic, 50:3 (2011), 211–225
7. E. A. Palyutin, “$P$-stable Abelian groups”, Algebra and Logic, 52:5 (2013), 404–421
8. E. A. Palyutin, “$P$-spectra of Abelian groups”, Algebra and Logic, 53:2 (2014), 140–165
9. E. A. Palyutin, “Theories of $P$-expansions of Abelian groups”, Algebra and Logic, 54:2 (2015), 183–187
10. E. A. Palyutin, “Totally $P$-stable Abelian groups”, Algebra and Logic, 54:4 (2015), 296–315
11. A. A. Stepanova, D. O. Ptakhov, “$P$-stable polygons”, Algebra and Logic, 56:4 (2017), 324–336
12. Yeshkeyev A.R., “About Central Types and the Cosemanticness of the Delta-Pm Fragment of the Jonsson Set”, Bull. Karaganda Univ-Math., 87:3 (2017), 51–58
13. D. O. Ptakhov, “Polygons with a (P, 1)-stable theory”, Algebra and Logic, 56:6 (2018), 473–478
14. Yeshkeyev A.R. Kassymetova M.T. Ulbrikht O.I., “Criterion For the Cosemanticness of the Abelian Groups in the Enriched Signature”, Bull. Karaganda Univ-Math., 89:1 (2018), 49–60
15. Yeshkeyev A.R., Omarova M.T., “Central Types of Convex Fragments of the Perfect Jonsson Theory”, Bull. Karaganda Univ-Math., 93:1 (2019), 95–101
16. A. A. Stepanova, A. I. Krasitskaya, “$P$-stabilnost nekotorykh klassov $S$-poligonov”, Sib. matem. zhurn., 62:2 (2021), 441–449
17. A. R. Yeshkeyev, M. T. Kassymetova, O. I. Ulbrikht, “Independence and simplicity in Jonsson theories with abstract geometry”, Sib. elektron. matem. izv., 18:1 (2021), 433–455
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