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Keywords:

  • generalized power series;
  • Hahn groups;
  • exponential extension;
  • exponential closure;
  • growth axioms;
  • morphisms of prelogarithmic fields;
  • MSC (2010) Primary: 06A05;
  • 12J10;
  • 12J15;
  • 12L12;
  • 13A18;
  • Secondary: 03C60;
  • 12F05;
  • 12F10;
  • 12F20

Abstract

We explain how the field of logarithmic-exponential series constructed in 20 and 21 embeds as an exponential field in any field of exponential-logarithmic series constructed in 9, 6, and 13. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Thequation image; the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions.