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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Exponentiation in power series fields
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by Franz-Viktor Kuhlmann, Salma Kuhlmann and Saharon Shelah PDF
Proc. Amer. Math. Soc. 125 (1997), 3177-3183 Request permission

Abstract:

We prove that for no nontrivial ordered abelian group $G$ does the ordered power series field $\mathbb {R}((G))$ admit an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field $k$, no exponential on $k((G))$ is compatible, that is, induces an exponential on $k$ through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.
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Additional Information
  • Franz-Viktor Kuhlmann
  • Affiliation: Mathematisches Institut der Universität Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany
  • Email: fvk@harmless.mathi.uni-heidelberg.de
  • Salma Kuhlmann
  • Affiliation: Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
  • MR Author ID: 293156
  • Saharon Shelah
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Email: shelah@sunrise.huji.ac.il
  • Received by editor(s): January 31, 1996
  • Received by editor(s) in revised form: May 18, 1996
  • Additional Notes: The second author was supported by a Deutsche Forschungsgemeinschaft fellowship. The third author was partially supported by the Edmund Landau Center for research in Mathematical Analysis, and supported by the Minerva Foundation (Germany). Publication number 601.
  • Communicated by: Andreas R. Blass
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 3177-3183
  • MSC (1991): Primary 12J15, 06A05; Secondary 12J25, 06F20
  • DOI: https://doi.org/10.1090/S0002-9939-97-03964-6
  • MathSciNet review: 1402868