Exponential fields and Conway's omega-map

TL;DR

This paper introduces omega-fields, a class of real closed fields inspired by surreal numbers, and demonstrates that certain omega-fields can be equipped with an exponential function, linking them to models of real exponential fields.

Contribution

The paper defines omega-fields and proves that bounded Hahn series omega-fields with real coefficients can be endowed with an exponential, extending the understanding of exponential models.

Findings

Omega-fields are real closed fields with value groups isomorphic to their additive reducts.

Bounded Hahn series omega-fields with real coefficients admit an exponential function.

The results connect omega-fields to models of the real exponential field.

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.