No CrossRef data available.
Article contents
Finitely axiomatizable strongly minimal groups
Published online by Cambridge University Press: 12 March 2014
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
- Copyright
- Copyright © Association for Symbolic Logic 2012
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
[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