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.

Overview

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

gives a clear statement and proof of the theorem. The papers of and together provide another proof. See also .

See also

* Archimedean group

References

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* {{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

Ordered groups
Theorems in group theory