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.¹

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

Gravett (1956) gives a clear statement and proof of the theorem.² The papers of Clifford (1954)³ and Hausner & Wendel (1952)⁴ together provide another proof.³⁴ See also Fuchs & Salce (2001, p. 62).