Comparison of Exponential-Logarithmic and Logarithmic-Exponential series

Abstract

We explain how the field of logarithmic-exponential series constructed in \cite{DMM1} and \cite {DMM2} embeds as an exponential field in any field of exponential-logarithmic series constructed in \cite{KK1}, \cite {K} and \cite {KS}. 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 Th$({\mathbb{R}}_{an,exp})$; the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions.

Cited by in corpus (1)

• On generalized series fields and exponential-logarithmic series fields with derivations