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- Hahn_embedding_theorem abstract "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.The theorem states that every linearly ordered abelian group G can be embedded as an ordered subgroup of the additive group ℝΩ endowed with a lexicographical order, where ℝ is the additive group of real numbers (with its standard order), and Ω is the set of Archimedean equivalence classes of G.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 ℝΩ 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. The papers of Clifford (1954) and Hausner & Wendel (1952) together provide another proof. See also Fuchs & Salce (2001, p. 62).".
- Hahn_embedding_theorem wikiPageID "1075596".
- Hahn_embedding_theorem wikiPageRevisionID "534054492".
- Hahn_embedding_theorem hasPhotoCollection Hahn_embedding_theorem.
- Hahn_embedding_theorem subject Category:Ordered_groups.
- Hahn_embedding_theorem subject Category:Theorems_in_algebra.
- Hahn_embedding_theorem type Abstraction100002137.
- Hahn_embedding_theorem type Communication100033020.
- Hahn_embedding_theorem type Group100031264.
- Hahn_embedding_theorem type Message106598915.
- Hahn_embedding_theorem type OrderedGroups.
- Hahn_embedding_theorem type Proposition106750804.
- Hahn_embedding_theorem type Statement106722453.
- Hahn_embedding_theorem type Theorem106752293.
- Hahn_embedding_theorem type TheoremsInAlgebra.
- Hahn_embedding_theorem comment "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.".
- Hahn_embedding_theorem label "Hahn embedding theorem".
- Hahn_embedding_theorem sameAs m.043ssr.
- Hahn_embedding_theorem sameAs Q5638886.
- Hahn_embedding_theorem sameAs Q5638886.
- Hahn_embedding_theorem sameAs Hahn_embedding_theorem.
- Hahn_embedding_theorem wasDerivedFrom Hahn_embedding_theorem?oldid=534054492.
- Hahn_embedding_theorem isPrimaryTopicOf Hahn_embedding_theorem.