The exponential-logarithmic equivalence classes of surreal numbers

TL;DR

This paper characterizes exponential equivalence classes within surreal numbers, providing recursive and sign sequence formulas for minimal representatives, advancing understanding of their algebraic and model-theoretic properties.

Contribution

It offers a complete description of exponential equivalence classes in surreal numbers, including explicit formulas for minimal representatives, extending prior foundational work.

Findings

Describes exponential equivalence classes in surreal numbers.

Provides recursive and sign sequence formulas for representatives.

Connects surreal numbers with model theory of real exponential fields.

Abstract

In his monograph, H. Gonshor showed that Conway's real closed field of surreal numbers carries an exponential and logarithmic map. Subsequently, L. van den Dries and P. Ehrlich showed that it is a model of the elementary theory of the field of real numbers with the exponential function. In this paper, we give a complete description of the exponential equivalence classes in the spirit of the classical Archimedean and multiplicative equivalence classes. This description is made in terms of a recursive formula as well as a sign sequence formula for the family of representatives of minimal length of these exponential classes.