SURREAL ORDERED EXPONENTIAL FIELDS

In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field 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 No, i.e. a subfield (K-subspace) of No that is an initial subtree of 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 (No, exp). These include all models of T(R-w, e(x)), where 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 No, which includes No itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field T-LE of logarithmic-exponential transseries into No is shown to be initial, as are the ordered exponential fields R((omega))(EL) and R <>.