Hahn embedding theorem

Short description: Description of linearly ordered groups

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 ${\mathbb{ℝ}}^{\mathrm{\Omega }}$ 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 ${\mathbb{ℝ}}^{\mathrm{\Omega }}$ 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 ${\mathbb{ℝ}}^{\mathrm{\Omega }}$ 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) together provide another proof. See also (Fuchs Salce).

See also

• Archimedean group

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 
• 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, From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, Kluwer Academic Publishers, pp. 165–213, http://www.ohio.edu/people/ehrlich/HahnNew.pdf, retrieved 2015-03-27 
• Hahn, H. (1907), "Über die nichtarchimedischen Größensysteme." (in German), Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse (Wien. Ber.) 116: 601–655