SURREAL ORDERED EXPONENTIAL FIELDS

Abstract In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\mathbf {No}}$ of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$ -vector space) to be isomorphic to an initial subfield ( $K$ -subspace) of ${\mathbf {No}}$ , i.e. a subfield ( $K$ -subspace) of ${\mathbf {No}}$ that is an initial subtree of ${\mathbf {No}}$ . In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of Schmeling’s conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $({\mathbf {No}}, \exp )$ . These include all models of $T({\mathbb R}_W, e^x)$ , where ${\mathbb R}_W$ is the reals expanded by a convergent Weierstrass system W. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of ${\mathbf {No}}$ , which includes ${\mathbf {No}}$ itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field ${\mathbb T}^{LE}$ of logarithmic-exponential transseries into ${\mathbf {No}}$ is shown to be initial, as are the ordered exponential fields ${\mathbb R}((\omega ))^{EL}$ and ${\mathbb R}\langle \langle \omega \rangle \rangle $ .