Comparison of Exponential-Logarithmic and Logarithmic-Exponential series

TL;DR

This paper compares two types of exponential-logarithmic series fields, showing how one embeds into the other and clarifying their fundamental differences in model-theoretic properties.

Contribution

It demonstrates the embedding of logarithmic-exponential series into exponential-logarithmic series fields and explains why the two constructions are inherently non-isomorphic.

Findings

Logarithmic-exponential series embed into exponential-logarithmic series fields.

No exponential-logarithmic series field embeds into logarithmic-exponential series.

The two constructions produce non-isomorphic models of Th(ℝ_{an, exp}).

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$(\R_{an, exp})$; the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions.