In mathematics, especially in the area of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...

dealing with ordered structures on

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

s, 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 In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...

of the additive group ℝ^Ω endowed with a lexicographical order, where ℝ is the additive group of real numbers (with its standard order), Ω is the set of ''Archimedean equivalence classes'' of ''G'', and ℝ^Ω is the set of all functions from Ω to ℝ which vanish outside a

well-ordered set In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-or ...

. Let 0 denote the

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...

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 Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...

, so ℝ^Ω 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 .

References

* * * * * * {{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, pages=977–982, doi-access=free Ordered groups Theorems in group theory