Finitely axiomatizable strongly minimal groups
Thomas Blossier and Elisabeth Bouscaren
Source: J. Symbolic Logic Volume 75, Issue 1
(2010), 25-50.
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.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1264433908
Digital Object Identifier: doi:10.2178/jsl/1264433908
Zentralblatt MATH identifier: 05681291
Mathematical Reviews number (MathSciNet): MR2605881
References
A. I. Abakumov, E. A. Paljutin, M. A. Taĭclin, and Ju. E. Šišmarev, Categorial quasivarieties, Algebra i Logika, vol. 11 (1972), pp. 3--38, 121.
Mathematical Reviews (MathSciNet): MR360407
G. Ahlbrandt and M. Ziegler, Quasi-finitely axiomatizable totally categorical theories, Annals of Pure and Applied Logic, vol. 30 (1986), no. 1, pp. 63--82.
Mathematical Reviews (MathSciNet): MR831437
Zentralblatt MATH: 0592.03018
Digital Object Identifier: doi:10.1016/0168-0072(86)90037-0
W. Baur, Elimination of quantifiers for modules, Israel Journal of Mathematics, vol. 25 (1976), no. 1-2, pp. 64--70.
Mathematical Reviews (MathSciNet): MR457194
Zentralblatt MATH: 0354.02043
Digital Object Identifier: doi:10.1007/BF02756561
S. Buechler, Essential Stability Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1996.
Mathematical Reviews (MathSciNet): MR1416106
Zentralblatt MATH: 0864.03025
--------, Vaught's conjecture for superstable theories of finite rank, Annals of Pure and Applied Logic, vol. 155 (2008), no. 3, pp. 135--172.
Mathematical Reviews (MathSciNet): MR2467429
Zentralblatt MATH: 1175.03019
Digital Object Identifier: doi:10.1016/j.apal.2008.03.004
G. Cherlin, L. Harrington, and A. H. Lachlan, $\aleph\sb 0$-categorical, $\aleph\sb 0$-stable structures, Annals of Pure and Applied Logic, vol. 28 (1985), no. 2, pp. 103--135.
Mathematical Reviews (MathSciNet): MR779159
Zentralblatt MATH: 0566.03022
Digital Object Identifier: doi:10.1016/0168-0072(85)90023-5
E. Fisher, Abelian structures, Ph.D. thesis, University of Yale, 1975.
H. B. Gute and K. K. Reuter, The last word on elimination of quantifiers in modules, Journal of Symbolic Logic, vol. 55 (1990), no. 2, pp. 670--673.
Mathematical Reviews (MathSciNet): MR1056380
Zentralblatt MATH: 0712.03018
Digital Object Identifier: doi:10.2307/2274656
JSTOR: links.jstor.org
W. Hodges, Model Theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.
Mathematical Reviews (MathSciNet): MR1221741
E. Hrushovski, Finitely axiomatizable $\aleph\sb 1$-categorical theories, Journal of Symbolic Logic, vol. 59 (1994), no. 3, pp. 838--844.
Mathematical Reviews (MathSciNet): MR1295972
Zentralblatt MATH: 0808.03015
Digital Object Identifier: doi:10.2307/2275911
JSTOR: links.jstor.org
E. Hrushovski and A. Pillay, 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.
Mathematical Reviews (MathSciNet): MR895647
Zentralblatt MATH: 0636.03028
A. A. Ivanov, The problem of finite axiomatizability for strongly minimal theories of graphs, Algebra i Logika, vol. 28 (1989), no. 3, pp. 280--297, 366.
Mathematical Reviews (MathSciNet): MR1066316
--------, Strongly minimal structures with disintegrated algebraic closure and structures of bounded valency, Proceedings of the Tenth Easter Conference on Model Theory (M. Weese and H. Wolter, editors), 1993.
Zentralblatt MATH: 0797.03033
S. Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002.
Mathematical Reviews (MathSciNet): MR1878556
J. A. Makowsky, 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.
Mathematical Reviews (MathSciNet): MR366647
D. Marker, Strongly minimal sets and geometry (Tutorial), Proceedings of the Logic Colloquium '95 (J. A. Makowsky, editor), Lecture Notes in Logic, vol. 11, Springer, Berlin, 1998, pp. 191--213.
Mathematical Reviews (MathSciNet): MR1678361
Zentralblatt MATH: 0895.03013
M. Morley, Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514--538.
Mathematical Reviews (MathSciNet): MR175782
Zentralblatt MATH: 0151.01101
Digital Object Identifier: doi:10.2307/1994188
JSTOR: links.jstor.org
E. A. Paljutin, Description of categorical quasivarieties, Algebra i Logika, vol. 14 (1975), no. 2, pp. 145--185, 240.
Mathematical Reviews (MathSciNet): MR398821
M. G. Peretjat'kin, Example of an $\omega \sb1$-categorical complete finitely axiomatizable theory, Algebra i Logika, vol. 19 (1980), no. 3, pp. 314--347, 382--383.
Mathematical Reviews (MathSciNet): MR609018
A. Pillay, Geometric Stability Theory, Oxford Logic Guides, vol. 32, The Clarendon Press Oxford University Press, New York, 1996.
Mathematical Reviews (MathSciNet): MR1429864
Zentralblatt MATH: 0871.03023
M. Prest, Model Theory and Modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988.
Mathematical Reviews (MathSciNet): MR933092
R. L. Vaught, Models of complete theories, Bulletin of the American Mathematical Society, vol. 69 (1963), pp. 299--313.
Mathematical Reviews (MathSciNet): MR147396
Zentralblatt MATH: 0112.00701
Digital Object Identifier: doi:10.1090/S0002-9904-1963-10903-9
Project Euclid: euclid.bams/1183525248
B. I. Zilber, 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.
Mathematical Reviews (MathSciNet): MR720383
--------, Uncountably Categorical Theories, Translations of Mathematical Monographs, vol. 117, American Mathematical Society, Providence, 1993.
Mathematical Reviews (MathSciNet): MR1206477
Zentralblatt MATH: 0785.03019
