In mathematics, an ordered exponential field is an

ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fie ...

together with a function which generalises the idea of exponential functions on the ordered field of real numbers.

Definition

An exponential

E

on an ordered field

K

is a strictly increasing

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

of the additive group of

K

onto the multiplicative group of positive elements of

K

. The ordered field

K\,

together with the additional function

E\,

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

a^x

where

a

is a real number greater than 1. One such function is the usual

exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...

, that is . The ordered field R equipped with this function gives the ordered real exponential field, denoted by . 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 Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...

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 number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surrea ...

\mathbf

admits an exponential which extends the exponential function exp on R. Since

\mathbf

does not have the

Archimedean property In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typi ...

, this is an example of a non-Archimedean ordered exponential field. * The ordered field of logarithmic-exponential transseries

\mathbb^

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

E

. For any formally exponential field

K

, one can choose an exponential

E

K

such that

1+1/n for some natural number n . Salma Kuhlmann, ''Ordered Exponential Fields'', Fields Institute Monographs, 12, (2000), p. 24.

Properties

* Every ordered exponential field

K

is ''root-closed'', i.e., every positive element of

K\,

has an

n

-th root for all positive integer

n

(or in other words the multiplicative group of positive elements of

K\,

divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

). This is so because

E\left(\fracE^(a)\right)^n=E(E^(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 number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...

s does not admit an exponential. This is so because an exponential

E

has to be of the form

E(x)=a^x\,

for some

1 in every formally exponential subfield K of the real numbers; however, E(\sqrt)=a^\sqrt is not algebraic if 1 is algebraic by the

Gelfond–Schneider theorem In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. History It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider. Statement : If ''a'' and ''b'' ar ...

. * 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 In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one ofte ...

structures. * The class of formally exponential fields is a pseudoelementary class. This is so since a field

K\,

is exponentially closed if and only if there is a surjective function

E_2\colon K\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.

Notes

References