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 groupss. It is named after Hans Hahn.

Overview

The theorem states that every linearly ordered abelian group G can be embedded as an ordered subgroup endowed with a lexicographical order is the additive group of real numbers (with its standard order), is the set of Archimedean equivalence classes of G is the set of all functions from 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 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