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On The Value Group Of The Transseries

Pacific Journal of Mathematics, ISSN: 1945-5844, Vol: 312, Issue: 2, Page: 335-354
2021
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Given a real closed field (K,C,., <), the possibility of defining an ordered exponential on it, that is, an ordered group isomorphism exp:(K,+,<)!.K>0; →(K,.<), is strictly connected to the properties of its natural valuation (i.e., the valuation whose valuation ring is given by the elements bounded in absolute value by some natural number). For example if an exponential exists, then the valuation group v.K/ is isomorphic to an additive complement of the valuation ring. In [Kuhlmann et al. 1997] it is shown that for ordered fields that are maximal with respect to their natural valuation this condition fails unless the value group is trivial (in which case K ⊆ R), so maximal nonarchimedean ordered fields do not admit an exponential. In [Kuhlmann and Shelah 2005] the same property is used, the other way around, to show that, given a regular uncountable cardinal κ, there is a nontrivial group M such that the field(Formula presented) of κ-bounded generalized series admits an exponential map.

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