Hahn embedding theorem explained
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]
Overview
The theorem states that every linearly ordered abelian group G can be embedded as an ordered subgroup of the additive group
endowed with a
lexicographical order, where
is the additive group of
real numbers (with its standard order), is the set of
Archimedean equivalence classes of
G, and
is the set of all
functions from to
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, so
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
Notes and References
- Web site: lo.logic - Hahn's Embedding Theorem and the oldest open question in set theory. 2021-01-28. MathOverflow.
