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 te ...

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 com ...

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 Embedded or embedding (alternatively imbedded or imbedding) may refer to: Science * Embedding, in mathematics, one instance of some mathematical object contained within another instance ** Graph embedding * Embedded generation, a distributed ge ...

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 In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of ...

, where ℝ is the additive group of

real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s (with its standard order), Ω is the set of ''Archimedean

equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...

es'' 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-o ...

. 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 s ...

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 number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

s ''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 of ...

, 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

it is a subgroup of the ordered additive group of the real numbers). gives a clear statement and

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...

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