Ordered exponential field
In mathematics, an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of real numbers.
Definition
An exponential [math]\displaystyle{ E }[/math] on an ordered field [math]\displaystyle{ K }[/math] is a strictly increasing isomorphism of the additive group of [math]\displaystyle{ K }[/math] onto the multiplicative group of positive elements of [math]\displaystyle{ K }[/math]. The ordered field [math]\displaystyle{ K\, }[/math] together with the additional function [math]\displaystyle{ E\, }[/math] is called an ordered exponential field.
Examples
- The canonical example for an ordered exponential field is the ordered field of real numbers R with any function of the form [math]\displaystyle{ a^x }[/math] where [math]\displaystyle{ a }[/math] is a real number greater than 1. One such function is the usual exponential function, that is E(x) = ex. The ordered field R equipped with this function gives the ordered real exponential field, denoted by Rexp. It was proved in the 1990s that Rexp is model complete, a result known as Wilkie's theorem. This result, when combined with Khovanskiĭ's theorem on pfaffian functions, proves that Rexp is also o-minimal.[1] Alfred Tarski posed the question of the decidability of Rexp and hence it is now known as Tarski's exponential function problem. It is known that if the real version of Schanuel's conjecture is true then Rexp is decidable.[2]
- The ordered field of surreal numbers [math]\displaystyle{ \mathbf{No} }[/math] admits an exponential which extends the exponential function exp on R. Since [math]\displaystyle{ \mathbf{No} }[/math] does not have the Archimedean property, this is an example of a non-Archimedean ordered exponential field.
- The ordered field of logarithmic-exponential transseries [math]\displaystyle{ \mathbb{T}^{LE} }[/math] is constructed specifically in a way such that it admits a canonical exponential.
Formally exponential fields
A formally exponential field, also called an exponentially closed field, is an ordered field that can be equipped with an exponential [math]\displaystyle{ E }[/math]. For any formally exponential field [math]\displaystyle{ K }[/math], one can choose an exponential [math]\displaystyle{ E }[/math] on [math]\displaystyle{ K }[/math] such that [math]\displaystyle{ 1+1/n\lt E(1)\lt n }[/math] for some natural number [math]\displaystyle{ n }[/math].[3]
Properties
- Every ordered exponential field [math]\displaystyle{ K }[/math] is root-closed, i.e., every positive element of [math]\displaystyle{ K\, }[/math] has an [math]\displaystyle{ n }[/math]-th root for all positive integer [math]\displaystyle{ n }[/math] (or in other words the multiplicative group of positive elements of [math]\displaystyle{ K\, }[/math] is divisible). This is so because [math]\displaystyle{ E\left(\frac{1}{n}E^{-1}(a)\right)^n=E(E^{-1}(a))=a }[/math] for all [math]\displaystyle{ a\gt 0 }[/math].
- Consequently, every ordered exponential field is a Euclidean field.
- Consequently, every ordered exponential field is an ordered Pythagorean field.
- Not every real-closed field is a formally exponential field, e.g., the field of real algebraic numbers does not admit an exponential. This is so because an exponential [math]\displaystyle{ E }[/math] has to be of the form [math]\displaystyle{ E(x)=a^x\, }[/math] for some [math]\displaystyle{ 1\lt a\in K }[/math] in every formally exponential subfield [math]\displaystyle{ K }[/math] of the real numbers; however, [math]\displaystyle{ E(\sqrt{2})=a^\sqrt{2} }[/math] is not algebraic if [math]\displaystyle{ 1\lt a }[/math] is algebraic by the Gelfond–Schneider theorem.
- Consequently, the class of formally exponential fields is not an elementary class since the field of real numbers and the field of real algebraic numbers are elementarily equivalent structures.
- The class of formally exponential fields is a pseudoelementary class. This is so since a field [math]\displaystyle{ K\, }[/math] is exponentially closed if and only if there is a surjective function [math]\displaystyle{ E_2\colon K\rightarrow K^+ }[/math] such that [math]\displaystyle{ E_2(x+y)=E_2(x)E_2(y) }[/math] and [math]\displaystyle{ E_2(1)=2 }[/math]; and these properties of [math]\displaystyle{ E_2 }[/math] are axiomatizable.
See also
Notes
- ↑ A.J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc., 9 (1996), pp. 1051–1094.
- ↑ A.J. Macintyre, A.J. Wilkie, On the decidability of the real exponential field, Kreisel 70th Birthday Volume, (2005).
- ↑ Salma Kuhlmann, Ordered Exponential Fields, Fields Institute Monographs, 12, (2000), p. 24.
References
- Alling, Norman L. (1962). "On Exponentially Closed Fields". Proceedings of the American Mathematical Society 13 (5): 706–711. doi:10.2307/2034159.
- Kuhlmann, Salma (2000), Ordered Exponential Fields, Fields Institute Monographs, 12, American Mathematical Society, doi:10.1090/fim/012, ISBN 0-8218-0943-1
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