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 on an ordered field is a strictly increasing
isomorphism of the additive group of onto the multiplicative group of positive elements of . The ordered field together with the additional function 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 where 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 admits an exponential which extends the exponential function exp on R. Since does not have the
Archimedean property, this is an example of a non-Archimedean ordered exponential field.
A formally exponential field, also called an exponentially closed field, is an ordered field that can be equipped with an exponential . For any formally exponential field , one can choose an exponential on such that
for some natural number .[3]
Properties
Every ordered exponential field is root-closed, i.e., every positive element of has an -th root for all positive integer (or in other words the multiplicative group of positive elements of is
divisible). This is so because for all .
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 has to be of the form for some in every formally exponential subfield of the real numbers; however, is not algebraic if 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 is exponentially closed if and only if there is a surjective function such that and ; and these properties of are axiomatizable.
^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.
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 on an ordered field is a strictly increasing
isomorphism of the additive group of onto the multiplicative group of positive elements of . The ordered field together with the additional function 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 where 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 admits an exponential which extends the exponential function exp on R. Since does not have the
Archimedean property, this is an example of a non-Archimedean ordered exponential field.
A formally exponential field, also called an exponentially closed field, is an ordered field that can be equipped with an exponential . For any formally exponential field , one can choose an exponential on such that
for some natural number .[3]
Properties
Every ordered exponential field is root-closed, i.e., every positive element of has an -th root for all positive integer (or in other words the multiplicative group of positive elements of is
divisible). This is so because for all .
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 has to be of the form for some in every formally exponential subfield of the real numbers; however, is not algebraic if 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 is exponentially closed if and only if there is a surjective function such that and ; and these properties of are axiomatizable.
^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.