Ordered exponential field

Short description: ordered field with a function generalizing the exponential function

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) = e^x. The ordered field R equipped with this function gives the ordered real exponential field, denoted by R_exp. It was proved in the 1990s that R_exp is model complete, a result known as Wilkie's theorem. This result, when combined with Khovanskiĭ's theorem on pfaffian functions, proves that R_exp is also o-minimal. Alfred Tarski posed the question of the decidability of R_exp 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 R_exp is decidable.
• 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].

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

• Exponential field

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