Ordered exponential field

An exponential $E$ on an ordered field $K$ is a strictly increasing isomorphism of the additive group of $K$ onto the multiplicative group of positive elements of $K$. The ordered field $K\backslash ,$ together with the additional function $E\backslash ,$ is called an ordered exponential field.

• The canonical example for an ordered exponential field is the ordered field of real numbers R with any function of the form $a^x$ where $a$ is a real number greater than 1. One such function is the usual exponential function, that is {{nowrap|1=E(x) = e^x}}. The ordered field R equipped with this function gives the ordered real exponential field, denoted by {{nowrap|1=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.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. 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.A.J. Macintyre, A.J. Wilkie, On the decidability of the real exponential field, Kreisel 70th Birthday Volume, (2005).
• The ordered field of surreal numbers $\backslash mathbf\{No\}$ admits an exponential which extends the exponential function exp on R. Since $\backslash mathbf\{No\}$ does not have the Archimedean property, this is an example of a non-Archimedean ordered exponential field.
• The ordered field of logarithmic-exponential transseries $\backslash mathbb\{T\}^\{LE\}$ is constructed specifically in a way such that it admits a canonical exponential.

A formally exponential field, also called an exponentially closed field, is an ordered field that can be equipped with an exponential $E$. For any formally exponential field $K$, one can choose an exponential $E$ on $K$ such that

Ordered Exponential Fields, Fields Institute Monographs, 12, (2000), p. 24.

• Every ordered exponential field $K$ is root-closed, i.e., every positive element of $K\backslash ,$ has an $n$-th root for all positive integer $n$ (or in other words the multiplicative group of positive elements of $K\backslash ,$ is divisible). This is so because $E\backslash left(\backslash frac\{1\}\{n\}E^\{-1\}(a)\backslash right)^n=E(E^\{-1\}(a))=a$ for all $a>0$.
• 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 $E$ has to be of the form $E(x)=a^x\backslash ,$ for some
• 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 $K\backslash ,$ is exponentially closed if and only if there is a surjective function $E\_2\backslash colon\; K\backslash rightarrow\; K^+$ such that $E\_2(x+y)=E\_2(x)E\_2(y)$ and $E\_2(1)=2$; and these properties of $E\_2$ are axiomatizable.

• Exponential field

{{Reflist}}

• {{cite journal | first=Norman L. | last=Alling | title=On Exponentially Closed Fields | journal=Proceedings of the American Mathematical Society | volume=13 | number=5 | year=1962 | pages=706–711 | zbl=0136.32201 | doi=10.2307/2034159| jstor=2034159 | doi-access=free }}
• {{Citation|last=Kuhlmann|first=Salma|title=Ordered Exponential Fields|series=Fields Institute Monographs|volume=12|publisher=American Mathematical Society|year=2000|isbn=0-8218-0943-1|mr=1760173|doi=10.1090/fim/012|doi-access=free}}

Category:Model theory

Category:Field (mathematics)

Category:Algebraic structures

Category:Exponentials