The Exponential-Logarithmic Equivalence Classes of Surreal Numbers

In his monograph [Gon86], 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 in [vdDE01] 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 (see Theorem 3.4) in the spirit of the classical Archimedean and multiplicative equivalence classes (see Theorem 2.4 and Proposition 2.5). 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. This result can be seen as a step towards proving that the field of surreal numbers No can be described as an exponential-logarithmic series field KEL (for some subfield of generalized series K of No) and a field of transseries T. Indeed, we conjecture that our representatives of the exponential classes are the fundamental monomials of K say the initial fundamental monomials in the sense of [KM11], or the logatomic elements (i.e. the monomials which remains monomials by taking iterated log’s) in the sense of [vdH06, Sch01]. Such a description would allow us to exploit these representations to introduce derivations on the surreals. Indeed, we know how to define derivations on exponential-logarithmic series fields [KM12, KM11] and on transseries fields [vdH97, Sch01]. In Section 2, we give a concise summary of the recursive definitions of the field operations on No, as well as the definition of the exponential exp and logarithmic log maps, and generalized epsilon numbers ǫNo. Our exposition is based on [Gon86]. Of particular interest to us is the analysis of certain equivalence relations on the surreal numbers. Conway [Con01, p 31-32] introduced and studied the ω-map to give a complete system ωNo (:= the image of No under this map) of representatives of the Archimedean additive equivalence relation. In [Gon86, Theorem 5.3], exploiting the convexity of the equivalence classes, Gonshor describes such a representative ωa as the unique surreal of minimal length in a given class. By a simple modification of their arguments, we describe a complete system ω No of representatives of the Archimedean multiplicative equivalence relation. In Section 3, we introduce and study what we call