# 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 linearly ordered abelian groups. It is named after Hans Hahn.

## Overview

The theorem states that every linearly ordered abelian group *G* can be embedded as an ordered subgroup of the additive group ℝ^{Ω} endowed with a lexicographical order, where ℝ is the additive group of real numbers (with its standard order), Ω is the set of *Archimedean equivalence classes* of *G*, and ℝ^{Ω} is the set of all functions from Ω to ℝ which vanish outside a well-ordered set.

Let 0 denote the identity element of *G*. For any nonzero element *g* of *G*, exactly one of the elements *g* or −*g* is greater than 0; denote this element by |*g*|. Two nonzero elements *g* and *h* of *G* are *Archimedean equivalent* if there exist natural numbers *N* and *M* such that *N*|*g*| > |h| and *M*|*h*| > |g|. Intuitively, this means that neither *g* nor *h* is "infinitesimal" with respect to the other. The group *G* is Archimedean if *all* nonzero elements are Archimedean-equivalent. In this case, Ω is a singleton, so ℝ^{Ω} is just the group of real numbers. Then Hahn's Embedding Theorem reduces to Hölder's theorem (which states that a linearly ordered abelian group is Archimedean if and only if it is a subgroup of the ordered additive group of the real numbers).

Gravett (1956) gives a clear statement and proof of the theorem. The papers of Clifford (1954) and Hausner & Wendel (1952) together provide another proof. See also Fuchs & Salce (2001, p. 62).

## See also

## References

- Fuchs, László; Salce, Luigi (2001),
*Modules over non-Noetherian domains*, Mathematical Surveys and Monographs,**84**, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR 1794715 - Ehrlich, Philip (1995), "Hahn's "Über die nichtarchimedischen Grössensysteme" and the Origins of the Modern Theory of Magnitudes and Numbers to Measure Them", in Hintikka, Jaakko (ed.),
*From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics*(PDF), Kluwer Academic Publishers, pp. 165–213 - Hahn, H. (1907), "Über die nichtarchimedischen Größensysteme.",
*Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse (Wien. Ber.)*(in German),**116**: 601–655 - Gravett, K. A. H. (1956), "Ordered Abelian Groups",
*The Quarterly Journal of Mathematics. Oxford. Second Series*,**7**: 57–63, doi:10.1093/qmath/7.1.57 - Clifford, A.H. (1954), "Note on Hahn's Theorem on Ordered Abelian Groups",
*Proceedings of the American Mathematical Society*,**5**(6): 860–863, doi:10.2307/2032549 - Hausner, M.; Wendel, J.G. (1952), "Ordered vector spaces",
*Proceedings of the American Mathematical Society*,**3**: 977–982, doi:10.1090/S0002-9939-1952-0052045-1