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Fusion over sublanguages

Published online by Cambridge University Press:  12 March 2014

Assaf Hasson  and
Martin Hils
Assaf Hasson
Affiliation:
The Mathematical Institute, 24–29 St. Giles, Oxford, OX1 3LB, UK
Martin Hils
Affiliation:
Institut für Mathematik, Humboldt-Universitätzu Berlin, D-10099 Berlin, Germany
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Abstract

Generalising Hrushovski's fusion technique we construct the free fusion of two strongly minimal theories T1. T2 intersecting in a totally categorical sub-theory T0. We show that if. e.g., T0 is the theory of infinite vector spaces over a finite field then the fusion theory Tω, exists, is complete and ω-stable of rank ω. We give a detailed geometrical analysis of Tω, proving that if both T1, T2 are 1-based then. Tω can be collapsed into a strongly minimal theory, if some additional technical conditions hold—all trivially satisfied if T0 is the theory of infinite vector spaces over a finite field .

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 2 , June 2006 , pp. 361 - 398
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Baldwin, J. T., Fundamentals of stability theory, Springer Verlag. 1988.CrossRefGoogle Scholar
[2]Baldwin, J. T., Notes on quasiminimality and excellence, preprint, 2004.CrossRefGoogle Scholar
[3]Baldwin, J. T. and Holland, K., Constructing ω-stable structures: Rank 2 fields, this Journal, vol. 65 (2000), pp. 371–391.Google Scholar
[4]Baldwin, J. T. and Holland, K., Constructing ω-stable structures: Computing rank, Fundamenta Mathematkae. vol. 170 (2001). pp. 1–20.CrossRefGoogle Scholar
[5]Baldwin, J. T., Constructing ω-stable structures: Model completeness, Annals of Pure and Applied Logic, vol. 125 (2004), pp. 159–172.CrossRefGoogle Scholar
[6]Baudisch, A., Martin-Pizarro, A., and Ziegler, M., Fields and colors, preprint. 2005.Google Scholar
[7]Baudisch, A., Martin-Pizarro, A., and Ziegler, M., Fusion over a vector space, preprint. 2005.CrossRefGoogle Scholar
[8]Cherlin, G.. Harrington, L., and Lachlan, A.. ℵ0-Categorical. ℵ0-stable structures, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 103–135.CrossRefGoogle Scholar
[9]Cherlin, G. and Hrushovski, E., Finite structures with few types, Annals of Mathematics Studies, vol. 152. Princeton University Press, 2003.Google Scholar
[10]Cohn, P. M., Skew field constructions, London Mathematical Society Lecture Note Series, vol. 27, Cambridge University Press, 1977.Google Scholar
[11]Gute, H. P. and Reuter, K. K.. The last word on elimination of quantifiers in modules, this JOurnal, vol. 55 (1990), pp. 670–673.Google Scholar
[12]Hasson, A.. Collapsing structure and a theory of envelopes, preprint. 2004.Google Scholar
[13]Hasson, A., Fusion over sublanguges revisited, preprint, 2006. (Available on author's webpage).Google Scholar
[14]Hils, M., La fusion au-dessus d'un sous-langage: le cas simple, preprint, 2004.Google Scholar
[15]Hils, M., Ph.D. thesis, in preperation.Google Scholar
[16]Holland, K., Model completeness of the new strongly minimal sets, this Journal, vol. 64 (1999), pp. 946–962.Google Scholar
[17]Hrushovski, E., Strongly minimal expansions of algebraically closed fields, Israel Journal of Mathematics, vol. 79 (1992). pp. 129–151.CrossRefGoogle Scholar
[18]Hrushovski, E., A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147–166.CrossRefGoogle Scholar
[19]Pillay, A.. Geometric stability theory, Clarendon Press, Oxford, 1996.CrossRefGoogle Scholar
[20]Poizat, B., Le carré de l'égalité, this Journal, vol. 64 (1999), pp. 1339–1355.Google Scholar
[21]Poizat, B., L'égalité au cube, this Journal, vol. 66 (2001), pp. 1647–1676.Google Scholar
[22]Zil'ber, B., Uncountably categorical theories. Translations of Mathematical Monographs, vol. 117, American Mathematical Society, Providence, R.I., 1993.CrossRefGoogle Scholar