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THERE ARE NO INTERMEDIATE STRUCTURES BETWEEN THE GROUP OF INTEGERS AND PRESBURGER ARITHMETIC

Published online by Cambridge University Press:  01 May 2018

GABRIEL CONANT
GABRIEL CONANT*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME NOTRE DAME, IN46656, USAE-mail:[email protected]
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Abstract

We show that if a first-order structure ${\cal M}$, with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then ${\cal M}$ must be interdefinable with (ℤ ,+,0) or (ℤ ,+,<,0).

Keywords

Primary 03C6403C0706F20Secondary 03C45Presburger arithmeticexpansions of the group of integersdp-minimality
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 187 - 207
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Alouf, E. and D’Elbée, C., A new dp-minimal expansion of the integers, 2017, arXiv:1707.07203 [math.LO].Google Scholar
Artin, M., Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991.Google Scholar
Aschenbrenner, M., Dolich, A., Haskell, D., Macpherson, D., and Starchenko, S., Vapnik-Chervonenkis density in some theories without the independence property, II. Notre Dame Journal of Formal Logic, vol. 54 (2013), no. 3–4, pp. 311–363.Google Scholar
Aschenbrenner, M., Dolich, A., Haskell, D., Macpherson, D., and Starchenko, S., Vapnik-Chervonenkis density in some theories without the independence property, I. Transactions of the American Mathematical Society, vol. 368 (2016), no. 8, pp. 5889–5949.Google Scholar
Bang, T., A solution of the “plank problem”. Proceedings of the American Mathematical Society, vol. 2 (1951), pp. 990–993.Google Scholar
Belegradek, O., Peterzil, Y., and Wagner, F., Quasi-o-minimal structures, this Journal, vol. 65 (2000), no. 3, pp. 1115–1132.Google Scholar
Cluckers, R., Presburger sets and p-minimal fields, this Journal, vol. 68 (2003), no. 1, pp. 153–162.Google Scholar
Conant, G. and Pillay, A., Stable groups and expansions of (ℤ,+,0). Preprint, 2016, arXiv:1601.05692.Google Scholar
Dolich, A. and Goodrick, J., Strong theories of ordered Abelian groups. Fundamenta Mathematicae, vol. 236 (2017), no. 3, pp. 269–296.CrossRefGoogle Scholar
Dolich, A., Goodrick, J., and Lippel, D., Dp-minimality: Basic facts and examples. Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 3, pp. 267–288.CrossRefGoogle Scholar
Kadets, V., Coverings by convex bodies and inscribed balls. Proceedings of the American Mathematical Society, vol. 133 (2005), no. 5, pp. 1491–1495.Google Scholar
Kaplan, I. and Shelah, S., Decidability and classification of the theory of integers with primes. Preprint, 2016, arXiv:1601.07099.Google Scholar
Marker, D., Semialgebraic expansions of C. Transactions of the American Mathematical Society, vol. 320 (1990), no. 2, pp. 581–592.Google Scholar
Marker, D., Model Theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002.Google Scholar
Michaux, C. and Villemaire, R., Presburger arithmetic and recognizability of sets of natural numbers by automata: New proofs of Cobham’s and Semenov’s theorems. Annals of Pure and Applied Logic, vol. 77 (1996), no. 3, pp. 251–277.Google Scholar
Palacín, D. and Sklinos, R., Superstable expansions of free abelian groups. Preprint, 2014, arXiv:1405.0568.Google Scholar
Pillay, A., Geometric Stability Theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996.CrossRefGoogle Scholar
Point, F., On decidable extensions of Presburger arithmetic: From A. Bertrand numeration systems to Pisot numbers, this Journal, vol. 65 (2000), no. 3, pp. 1347–1374.Google Scholar
Poizat, B., Supergénérix. Journal of Algebra, vol. 404 (2014), pp. 240–270, À la mémoire d’Éric Jaligot. [In memoriam Éric Jaligot].CrossRefGoogle Scholar
Presburger, M., Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, Comptes Rendus du I congrès de Mathématiciens des Pays Slaves, Warszawa, 1929, pp. 92–101.Google Scholar
Prest, M., Model Theory and Modules, Handbook of Algebra, vol. 3, North-Holland, Amsterdam, 2003, pp. 227–253.Google Scholar
Simon, P., A Guide to NIP Theories, Lecture Notes in Logic, vol. 44, Association for Symbolic Logic, Chicago, IL; Cambridge Scientific Publishers, Cambridge, 2015.CrossRefGoogle Scholar
Tarski, A., Uwagi o stopniu równoważności wielokatów. Parametr, vol. 2 (1931–1932), pp. 310–314.Google Scholar