Hahn embedding theorem

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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.^[1]

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

• Archimedean group

• 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, 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, archived from the original (PDF) on 2014-10-27, retrieved 2015-03-27
• 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, 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, JSTOR 2032549
• Hausner, M.; Wendel, J.G. (1952), "Ordered vector spaces", Proceedings of the American Mathematical Society, 3 (6): 977–982, doi:10.1090/S0002-9939-1952-0052045-1