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Finitely axiomatizable strongly minimal groups

Published online by Cambridge University Press:  12 March 2014

Thomas Blossier  and
Elisabeth Bouscaren
Thomas Blossier
Affiliation:
Université Lyon 1, Institut Camille Jordan Umr 5208 CNRS, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France, E-mail: [email protected]
Elisabeth Bouscaren
Affiliation:
CNRS, Univ. Paris-Sud 11, Département de Mathématiques d' Orsay, Orsay Cedex, F-91405, France, E-mail: [email protected]
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Abstract

We show that if G is a strongly minimal finitely axiomatizable group, the division ring of quasi-endomorphisms of G must be an infinite finitely presented ring.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 1 , March 2010 , pp. 25 - 50
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] Abakumov, A. I., Paljutin, E. A., Taĭclin, M. A., and Šišmarev, Ju. E., Categorial quasivarieties, Algebra i Logika, vol. 11 (1972), pp. 3–38, 121.Google Scholar
[2] Ahlbrandt, G. and Ziegler, M., Quasi-finitely axiomatizable totally categorical theories, Annals of Pure and Applied Logic, vol. 30 (1986), no. 1, pp. 63–82.Google Scholar
[3] Baur, W., Elimination of quantifiers for modules, Israel Journal of Mathematics, vol. 25 (1976), no. 1–2, pp. 64–70.Google Scholar
[4] Buechler, S., Essential Stability Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1996.Google Scholar
[5] Buechler, S., Vaught's conjecture for superstable theories of finite rank, Annals of Pure and Applied Logic, vol. 155 (2008), no. 3, pp. 135–172.Google Scholar
[6] Cherlin, G., Harrington, L., and Lachlan, A. H., ℵ0-categorical, ℵ0-stable structures, Annals of Pure and Applied Logic, vol. 28 (1985), no. 2, pp. 103–135.Google Scholar
[7] Fisher, E., Abelian structures, Ph.D. thesis, University of Yale, 1975.Google Scholar
[8] Gute, H. B. and Reuter, K. K., The last word on elimination of quantifiers in modules, this Journal, vol. 55 (1990), no. 2, pp. 670–673.Google Scholar
[9] Hodges, W., Model Theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.Google Scholar
[10] Hrushovski, E., Finitely axiomatizable ℵ1-categorical theories, this Journal, vol. 59 (1994), no. 3, pp. 838–844.Google Scholar
[11] Hrushovski, E. and Pillay, A., Weakly normal groups, Logic Colloquium '85 (Orsay, 1985), Studies in Logic and the Foundations of Mathematics, vol. 122, North-Holland, Amsterdam, 1987, pp. 233–244.Google Scholar
[12] Ivanov, A. A., The problem of finite axiomatizability for strongly minimal theories of graphs, Algebrai Logika, vol. 28 (1989), no. 3, pp. 280–297, 366.Google Scholar
[13] Ivanov, A. A., Strongly minimal structures with disintegrated algebraic closure and structures of bounded valency, Proceedings of the Tenth Easter Conference on Model Theory (Weese, M. and Wolter, H., editors), 1993.Google Scholar
[14] Lang, S., Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002.Google Scholar
[15] Makowsky, J. A., On some conjectures connected with complete sentences, Fundamenta Mathematicae, vol. 81 (1974), pp. 193–202, Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, III.Google Scholar
[16] Marker, D., Strongly minimal sets and geometry (Tutorial), Proceedings of the Logic Colloquium '95 (Makowsky, J. A., editor), Lecture Notes in Logic, vol. 11, Springer, Berlin, 1998, pp. 191–213.Google Scholar
[17] Morley, M., Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514–538.Google Scholar
[18] Paljutin, E. A., Description of categorical quasivarieties, Algebra i Logika, vol. 14 (1975), no. 2, pp. 145–185, 240.Google Scholar
[19] Peretjat′kin, M. G., Example of an ω1-categorical complete finitely axiomatizable theory, Algebra i Logika, vol. 19 (1980), no. 3, pp. 314–347, 382–383.Google Scholar
[20] Pillay, A., Geometric Stability Theory, Oxford Logic Guides, vol. 32, The Clarendon Press Oxford University Press, New York, 1996.Google Scholar
[21] Prest, M., Model Theory and Modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988.Google Scholar
[22] Vaught, R. L., Models of complete theories, Bulletin of the American Mathematical Society, vol. 69 (1963), pp. 299–313.Google Scholar
[23] Zilber, B. I., Solution of the problem of finite axiomatizability for theories that are categorical in all infinite powers, Investigations in Theoretical Programming, 1981, (Russian), pp. 69–74.Google Scholar
[24] Zilber, B. I., Uncountably Categorical Theories, Translations of Mathematical Monographs, vol. 117, American Mathematical Society, Providence, 1993.Google Scholar