Ordered exponential field

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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.

Contents

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

Formally exponential fields

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

See also

Notes

  1. 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.
  2. A.J. Macintyre, A.J. Wilkie, On the decidability of the real exponential field, Kreisel 70th Birthday Volume, (2005).
  3. Salma Kuhlmann, Ordered Exponential Fields, Fields Institute Monographs, 12, (2000), p. 24.

References