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SURREAL ORDERED EXPONENTIAL FIELDS
Published online by Cambridge University Press: 13 August 2021
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 $
.
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- © Association for Symbolic Logic 2021
Footnotes
The in-text reference citations have been corrected. An erratum detailing this update has also been published (doi: 10.1017/jsl.2022.5).
References
REFERENCES
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