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.

The theorem states: Any 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), and Ω is the set of Archimedean equivalence classes of $G$.

Let $0$ denote the identity element of $G$. For any nonzero $g$∈$G$, exactly one of the elements $g$ or $-g$ is greater than $0$; denote this element by $|g|$. Two nonzero elements $g,h$∈$G$ are Archimedean equivalent if there exist natural numbers $N,M$∈ℕ such that $N |g| > |h|$ and $M |h| > |g|$. (Heuristically: 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).

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, MR1794715 
• Hahn, H. (1907), "Über die nichtarchimedischen Größensysteme." (in German), Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse (Wien. Ber.) 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 
• 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