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.{{Cite web|title=lo.logic - Hahn's Embedding Theorem and the oldest open question in set theory|url=https://mathoverflow.net/questions/128935/hahns-embedding-theorem-and-the-oldest-open-question-in-set-theory|access-date=2021-01-28|website=MathOverflow}}

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), {{math|Ω}} is the set of Archimedean equivalence classes of G, and $\mathbb{R}^\Omega$ is the set of all functions from {{math|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, {{math|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).

{{harvtxt|Gravett|1956}} gives a clear statement and proof of the theorem. The papers of {{harvtxt|Clifford|1954}} and {{harvtxt|Hausner|Wendel|1952}} together provide another proof. See also {{harvtxt|Fuchs|Salce|2001|p=62}}.

References

{{Citation | last1=Fuchs | first1=László | last2=Salce | first2=Luigi | title=Modules over non-Noetherian domains | publisher=American Mathematical Society | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-1963-0 |mr=1794715 | year=2001 | volume=84}}
{{Citation | last1=Ehrlich | first1=Philip | url=http://www.ohio.edu/people/ehrlich/HahnNew.pdf | chapter=Hahn's "Über die nichtarchimedischen Grössensysteme" and the Origins of the Modern Theory of Magnitudes and Numbers to Measure Them | title=From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics |editor-first=Jaakko |editor-last=Hintikka | publisher= Kluwer Academic Publishers |year=1995|pages=165–213}}
{{Citation | last1=Hahn | first1=H. | author1-link=Hans Hahn (mathematician) | title=Über die nichtarchimedischen Größensysteme. | language=German | year=1907 | journal=Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse (Wien. Ber.) | volume=116 | pages=601–655}}
{{Citation | doi=10.1093/qmath/7.1.57 | last1=Gravett | first1=K. A. H.| title=Ordered Abelian Groups| journal=The Quarterly Journal of Mathematics |series=Second Series| volume=7| year=1956| pages=57–63}}
{{Citation | last1=Clifford | first1=A.H. | title= Note on Hahn's Theorem on Ordered Abelian Groups| journal=Proceedings of the American Mathematical Society| volume=5| issue=6| year=1954| pages=860–863 | doi=10.2307/2032549| jstor=2032549 }}
{{Citation | doi=10.1090/S0002-9939-1952-0052045-1 | last1=Hausner| first1=M. | last2=Wendel| first2=J.G.| title=Ordered vector spaces| journal=Proceedings of the American Mathematical Society| volume=3| year=1952| issue=6| pages=977–982| doi-access=free}}

Category:Ordered groups

Category:Theorems in group theory