Hahn embedding theorem
Template:Short description Template:More footnotes 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
OverviewEdit
The theorem states that every linearly ordered abelian group G can be embedded as an ordered subgroup of the additive group <math>\mathbb{R}^\Omega</math> endowed with a lexicographical order, where <math>\mathbb{R}</math> is the additive group of real numbers (with its standard order), Template:Math is the set of Archimedean equivalence classes of G, and <math>\mathbb{R}^\Omega</math> is the set of all functions from Template:Math to <math>\mathbb{R}</math> 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, Template:Math is a singleton, so <math>\mathbb{R}^\Omega</math> 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).
Template:Harvtxt gives a clear statement and proof of the theorem. The papers of Template:Harvtxt and Template:Harvtxt together provide another proof. See also Template:Harvtxt.