Exponential fields and Conway’s omega-map

Berarducci, Alessandro; Kuhlmann, Salma; Mantova, Vincenzo; Matusinski, Mickaël

doi:10.1090/proc/14577

Exponential fields and Conway’s omega-map
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by Alessandro Berarducci, Salma Kuhlmann, Vincenzo Mantova and Mickaël Matusinski

Proc. Amer. Math. Soc. 151 (2023), 2655-2669

DOI: https://doi.org/10.1090/proc/14577

Published electronically: March 21, 2023

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Abstract:

Inspired by Conway’s surreal numbers, we study real closed fields whose value group is isomorphic to the additive reduct of the field. We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. We also consider relative versions with more general coefficient fields.

References

Alessandro Berarducci and Vincenzo Mantova, Surreal numbers, derivations and transseries, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 2, 339–390. MR 3760298, DOI 10.4171/JEMS/769
Alessandro Berarducci and Vincenzo Mantova, Transseries as germs of surreal functions, Trans. Amer. Math. Soc. 371 (2019), no. 5, 3549–3592. MR 3896122, DOI 10.1090/tran/7428
J. H. Conway, On numbers and games, London Mathematical Society Monographs, No. 6, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0450066
Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
Lou van den Dries, Angus Macintyre, and David Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), no. 1, 183–205. MR 1289495, DOI 10.2307/2118545
Lou van den Dries and Philip Ehrlich, Fields of surreal numbers and exponentiation. Fund. Math. (2001) 167, no. 2, 173–188, and Erratum, Fund. Math. (2001) 168, no. 3, 295–297.
Harry Gonshor, An introduction to the theory of surreal numbers, London Mathematical Society Lecture Note Series, vol. 110, Cambridge University Press, Cambridge, 1986. MR 872856, DOI 10.1017/CBO9780511629143
Hans Hahn, Über die nichtarchimedischen Grössenszsteme, Sitz. K. Akad Wiss. Math. Nat. Kl. 116 (1907), no. IIa, 601–655.
Irving Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321. MR 6161
Franz-Viktor Kuhlmann, Salma Kuhlmann, and Saharon Shelah, Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3177–3183. MR 1402868, DOI 10.1090/S0002-9939-97-03964-6
Salma Kuhlmann, Ordered exponential fields, Fields Institute Monographs, vol. 12, American Mathematical Society, Providence, RI, 2000. MR 1760173, DOI 10.1090/fim/012
Salma Kuhlmann and Saharon Shelah, $\kappa$-bounded exponential-logarithmic power series fields, Ann. Pure Appl. Logic 136 (2005), no. 3, 284–296. MR 2169687, DOI 10.1016/j.apal.2005.04.001
M.-H. Mourgues and J. P. Ressayre, Every real closed field has an integer part, J. Symbolic Logic 58 (1993), no. 2, 641–647. MR 1233929, DOI 10.2307/2275224
B. H. Neumann, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202–252. MR 32593, DOI 10.1090/S0002-9947-1949-0032593-5
J.-P. Ressayre, Integer parts of real closed exponential fields (extended abstract), Arithmetic, proof theory, and computational complexity (Prague, 1991) Oxford Logic Guides, vol. 23, Oxford Univ. Press, New York, 1993, pp. 278–288. MR 1236467