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.^[1]

Overview

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

References

^ "lo.logic - Hahn's Embedding Theorem and the oldest open question in set theory". MathOverflow. Retrieved 2021-01-28.

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
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
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

[1] "lo.logic - Hahn's Embedding Theorem and the oldest open question in set theory". MathOverflow. Retrieved 2021-01-28.

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