## Notre Dame Journal of Formal Logic

### Lascar Types and Lascar Automorphisms in Abstract Elementary Classes

#### Abstract

We study Lascar strong types and Galois types and especially their relation to notions of type which have finite character. We define a notion of a strong type with finite character, the so-called Lascar type. We show that this notion is stronger than Galois type over countable sets in simple and superstable finitary AECs. Furthermore, we give an example where the Galois type itself does not have finite character in such a class.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 52, Number 1 (2011), 39-54.

Dates
First available in Project Euclid: 13 December 2010

https://projecteuclid.org/euclid.ndjfl/1292249609

Digital Object Identifier
doi:10.1215/00294527-2010-035

Mathematical Reviews number (MathSciNet)
MR2747161

Zentralblatt MATH identifier
1233.03038

#### Citation

Hyttinen, Tapani; Kesälä, Meeri. Lascar Types and Lascar Automorphisms in Abstract Elementary Classes. Notre Dame J. Formal Logic 52 (2011), no. 1, 39--54. doi:10.1215/00294527-2010-035. https://projecteuclid.org/euclid.ndjfl/1292249609

#### References

• [1] Baldwin, J. T., Categoricity, vol. 50 of University Lecture Series, American Mathematical Society, Providence, 2009.
• [2] Baldwin, J. T., P. C. Eklof, and J. Trlifaj, "${}^\perp N$" as an abstract elementary class", Annals of Pure and Applied Logic, vol. 149 (2007), pp. 25--39.
• [3] Baldwin, J. T., and A. Kolesnikov, "Categoricity, amalgamation, and tameness", Israel Journal of Mathematics, vol. 170 (2009), pp. 411--43.
• [4] Baldwin, J. T., A. Kolesnikov, and S. Shelah, "The amalgamation spectrum", The Journal of Symbolic Logic, vol. 74 (2009), pp. 914--28.
• [5] Bays, M., and B. Zilber, "Covers of multiplicative groups of algebraically closed fields of arbitrary characteristic". Available at arXiv:0704.3561v3 [math.LO].
• [6] Buechler, S., and O. Lessmann, "Simple homogeneous models", Journal of the American Mathematical Society, vol. 16 (2003), pp. 91--121.
• [7] Hrushovski, E., "Almost orthogonal regular types (Stability in Model Theory, II" (Trento, 1987), Annals of Pure and Applied Logic, vol. 45 (1989), pp. 139--55.
• [8] Hyttinen, T., and M. Kesälä, Categoricity transfer in simple finitary abstract elementary classes,'' Submitted in 2008. Available at http://mathstat.helsinki.fi/logic/people/meeri.kesala.html.
• [9] Hyttinen, T., and M. Kesälä, "Independence in finitary abstract elementary classes", Annals of Pure and Applied Logic, vol. 143 (2006), pp. 103--38.
• [10] Hyttinen, T., and M. Kesälä, "Interpreting groups and fields in simple, finitary AEC"s, (2010). Institut Mittag-Leffler preprint series 2009/2010. Available at http://www.mittag-leffler.se/preprints.
• [11] Hyttinen, T., and M. Kesälä, "Superstability in simple finitary AEC"s, Fundamenta Mathematicae, vol. 195 (2007), pp. 221--68.
• [12] Hyttinen, T., and O. Lessmann, "A rank for the class of elementary submodels of a superstable homogeneous model", The Journal of Symbolic Logic, vol. 67 (2002), pp. 1469--82.
• [13] Hyttinen, T., and O. Lessmann, "Simplicity and uncountable categoricity in excellent classes", Annals of Pure and Applied Logic, vol. 139 (2006), pp. 110--37.
• [14] Hyttinen, T., O. Lessmann, and S. Shelah, "Interpreting groups and fields in some nonelementary classes", Journal of Mathematical Logic, vol. 5 (2005), pp. 1--47.
• [15] Hyttinen, T., and S. Shelah, "Strong splitting in stable homogeneous models", Annals of Pure and Applied Logic, vol. 103 (2000), pp. 201--28.
• [16] Keisler, H. J., Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers, vol. 62 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, 1971.
• [17] Kesälä, M., Finitary Abstract Elementary Classes, Ph.D. thesis, University of Helsinki, Department of Mathematics and Statistics, 2006.
• [18] Kueker, D. W., "Abstract elementary classes and infinitary logics", Annals of Pure and Applied Logic, vol. 156 (2008), pp. 274--86.
• [19] Lascar, D., "The group of automorphisms of a relational saturated structure", pp. 225--36 in Finite and Infinite Combinatorics in Sets and Logic (Banff, 1991), vol. 411 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Kluwer Academic Publishers, Dordrecht, 1993.
• [20] Lascar, D., "On the category of models of a complete theory", The Journal of Symbolic Logic, vol. 47 (1982), pp. 249--66.
• [21] Shelah, S., "Classification of nonelementary classes. II". Abstract elementary classes, pp. 419--97 in Classification Theory (Proceedings, Chicago, 1985), edited by J. T. Baldwin, vol. 1292 of Lecture Notes in Mathematics, Springer, Berlin, 1987.
• [22] Shelah, S., Classification Theory for Elementary Abstract Classes, vol. 18 of Studies in Logic (London), College Publications, London, 2009. Mathematical Logic and Foundations.
• [23] Trlifaj, J., "Abstract elementary classes induced by tilting and cotilting modules have finite character", Proceedings of the American Mathematical Society, vol. 137 (2009), pp. 1127--33.