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Preprint Number 1730

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1730. Philip Ehrlich and Elliot Kaplan
Surreal ordered exponential fields

Submission date: 18 February 2020


In [15], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field No of surreal numbers was brought to the fore 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 to [15], piggybacking on the just-said results, analogous results are established for ordered exponential fields. 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. This uses the precursory result that trigonometric-exponential initial subfields of No and trigonometric ordered initial subfields of No, more generally, admit canonical sine and cosine functions. This is shown to apply to the members of a distinguished family of initial exponential subfields of No, to the image of the canonical map of the ordered exponential field 𝕋 of transseries into No, which is shown to be initial, and to the ordered exponential fields ℝ((ω))^{EL} and ℝ<<ω>>, which are likewise shown to be initial.

Mathematics Subject Classification: Primary 06A05, 03C64, Secondary 12J15, 06F20, 06F25

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Full text arXiv 2002.07739: pdf, ps.

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