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A P-MINIMAL STRUCTURE WITHOUT DEFINABLE SKOLEM FUNCTIONS

Published online by Cambridge University Press:  15 May 2017

PABLO CUBIDES KOVACSICS  and
KIEN HUU NGUYEN
PABLO CUBIDES KOVACSICS
Affiliation:
LABORATOIRE DE MATHÉMATIQUES NICOLAS ORESME UNIVERSITÉ DE CAEN CNRS UMR 6139 UNIVERSITÉ DE CAEN BP 5186 14032 CAEN CEDEX FRANCE E-mail: [email protected]
KIEN HUU NGUYEN
Affiliation:
LABORATOIRE PAUL PAINLEVÉ UNIVERSITÉ DE LILLE 1 CNRS U.M.R. 8524 59655 VILLENEUVE D’ASCQ CEDEX FRANCE and DEPARTMENT OF MATHEMATICS HANOI NATIONAL UNIVERSITY OF EDUCATION 136 XUANTHUY STR., CAU GIAY HANOI VIETNAM E-mail: [email protected]
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Abstract

We show there are intermediate P-minimal structures between the semialgebraic and subanalytic languages which do not have definable Skolem functions. As a consequence, by a result of Mourgues, this shows there are P-minimal structures which do not admit classical cell decomposition.

Keywords

12J1203C9912J25P-minimalitySkolem functionsp-adically closed fieldscell decompositioncell preparation
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 2 , June 2017 , pp. 778 - 786
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

Basarab, S. A., Relative elimination of quantifiers for henselian valued fields . Annals of Pure and Applied Logic, vol. 53 (1991), no. 1, pp. 51–74.Google Scholar
Cluckers, R., Analytic p-adic cell decomposition and integrals . Transactions of the American Mathematical Society, vol. 356 (2003), no. 4, pp. 1489–1499.CrossRefGoogle Scholar
Cluckers, R. and Lipshitz, L., Fields with analytic structure . Journal of the European Mathematical Society, vol. 13 (2011), pp. 1147–1223.Google Scholar
Cubides Kovacsics, P. and Leenknegt, E., Integration and cell decomposition in P-minimal structures, this Journal, vol. 81 (2016), no. 3, pp. 1124–1141.Google Scholar
Cubides Kovacsics, P., Leenknegt, E., and Darnière, L., Topological cell decomposition and dimension theory in p-minimal fields, this Journal, vol. 82 (2017), no. 1, pp. 347–358.Google Scholar
Darnière, L. and Halupczok, I., Cell decomposition and classification of definable sets in p-optimal fields, this Journal, vol. 82 (2017), no. 1, pp. 120–136.Google Scholar
Denef, J., p-adic semi-algebraic sets and cell decomposition . Crelles Journal, vol. 369 (1986), pp. 154–166.Google Scholar
Flenner, J., Relative decidability and definability in henselian valued fields, this Journal, vol. 76 (2011), no. 4, pp. 1240–1260.Google Scholar
Haskell, D. and Macpherson, D., A version of o-minimality for the p-adics, this Journal, vol. 62 (1997), no. 4, pp. 1075–1092.Google Scholar
Kaplansky, I., Maximal fields with valuations . Duke Mathematical Journal, vol. 9 (1942), no. 2, pp. 303–321.Google Scholar
Kuhlmann, F.-V., Quantifier elimination for henselian fields relative to additive and multiplicative congruences . Israel Journal of Mathematics, vol. 85 (1994), no. 1, pp. 277–306.CrossRefGoogle Scholar
MacIntyre, A., On definable subsets of p-adic fields, this Journal, vol. 41 (1976), no. 3, pp. 605–610.Google Scholar
Mourgues, M.-H., Cell decomposition for P-minimal fields . Mathematical Logic Quarterly, vol. 55 (2009), no. 5, pp. 487–492.Google Scholar
Pas, J., Uniform p-adic cell decomposition and local zeta functions . Crelles Journal, vol. 399 (1989), pp. 137–172.Google Scholar
Prestel, A. and Roquette, P., Formally p-adic Fields, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1984.Google Scholar
Rideau, S., Some properties of analytic difference valued fields . Journal of the Institute of Mathematics of Jussieu, vol. 16 (2017), no. 3, pp. 447–499.Google Scholar
Scanlon, T., Quantifier Elimination for the Relative Frobenius, Valuation theory and its applications, vol. II, Saskatoon, SK, 1999, Fields Institute Communications, vol. 33, American Mathematical Society, Providence, RI, 2003, pp. 323–352.Google Scholar
van den Dries, L, Haskell, D., and Macpherson, D., One-dimensional p-adic subanalytic sets . Journal of the London Mathematical Society, vol. 59 (1999), pp. 1–20.Google Scholar