Hahn embedding theorem
In mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all totally ordered abelian groups.
The theorem states that every totally ordered abelian group can be embedded as an ordered subgroup of the Hahn product of copies of the ordered additive group of the real numbers, (Fuchs & Salce 2001, p. 62).
It is therefore a generalization of Hölder's theorem that a totally ordered abelian group is Archimedian if and only if it is a subgroup of the ordered additive group of the real numbers.
References
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- Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR1794715
- Hahn, H. (1907), "Über die nichtarchimedischen Größ ensysteme.", Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse (Wien. Ber.) (in German), 116: 601–655