The Schröder-Bernstein property for $a$-saturated models
HTML articles powered by AMS MathViewer
- by John Goodrick and Michael C. Laskowski
- Proc. Amer. Math. Soc. 142 (2014), 1013-1023
- DOI: https://doi.org/10.1090/S0002-9939-2013-11844-7
- Published electronically: December 10, 2013
- PDF | Request permission
Abstract:
A first-order theory $T$ has the Schröder-Bernstein (SB) property if any pair of elementarily bi-embeddable models are isomorphic. We prove that $T$ has an expansion by constants with the SB property if and only if $T$ is superstable and non-multidimensional. We also prove that among superstable theories $T$, the class of $a$-saturated models of $T$ has the SB property if and only if $T$ has no nomadic types.References
- John Goodrick, When are elementarily bi-embeddable structures isomorphic?, Ph.D. thesis under Thomas Scanlon at UC Berkeley, 2007.
- John Goodrick and Michael C. Laskowski, The Schröder-Bernstein property for weakly minimal theories, Israel J. Math. 188 (2012), 91–110. MR 2897724, DOI 10.1007/s11856-011-0093-6
- Bradd T. Hart, Alistair H. Lachlan, and Matthew A. Valeriote (eds.), Algebraic model theory, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 496, Kluwer Academic Publishers Group, Dordrecht, 1997. MR 1481436, DOI 10.1007/978-94-015-8923-9
- Michael Morley, Countable models of $\aleph _{1}$-categorical theories, Israel J. Math. 5 (1967), 65–72. MR 219405, DOI 10.1007/BF02771623
- T. A. Nurmagambetov, The mutual elementary embeddability of models, Theory of algebraic structures (Russian), Karagand. Gos. Univ., Karaganda, 1985, pp. 109–115 (Russian). MR 922949
- T. A. Nurmagambetov, Characterization of $\omega$-stable theories with a bounded number of dimensions, Algebra i Logika 28 (1989), no. 5, 584–596, 611 (Russian); English transl., Algebra and Logic 28 (1989), no. 5, 388–396 (1990). MR 1087574, DOI 10.1007/BF01979199
- Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
- Mike Prest, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988. MR 933092, DOI 10.1017/CBO9780511600562
- S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551
- W. Szmielew, Elementary properties of Abelian groups, Fund. Math. 41 (1955), 203–271. MR 72131, DOI 10.4064/fm-41-2-203-271
Bibliographic Information
- John Goodrick
- Affiliation: Departamento de Matemáticas, Universidad de los Andes, Carrera 1 No. 18A-10, 111711 Bogotá, Colombia
- Email: jr.goodrick427@uniandes.edu.co
- Michael C. Laskowski
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- Email: mcl@math.umd.edu
- Received by editor(s): February 29, 2012
- Received by editor(s) in revised form: April 15, 2012
- Published electronically: December 10, 2013
- Additional Notes: The first author was partially supported by a grant from the Facultad de Ciencias, Universidad de los Andes, and by a travel grant from MSRI to attend the BIRS workshop on Neostability Theory (January 29–February 4, 2012).
The second author was partially supported by NSF grant DMS-0901336. - Communicated by: Julia Knight
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1013-1023
- MSC (2010): Primary 03C45
- DOI: https://doi.org/10.1090/S0002-9939-2013-11844-7
- MathSciNet review: 3148535