Math 2.0

Order, lattices, ordered algebraic structures Ordered algebraic structures Theorem

Hahn embedding theorem

Builds on Linearly ordered groupArchimedean equivalence classesAbelian groupWell-orderGroup homomorphismReal numbers

Statement

Let GG be an abelian group with a total order that makes it a linearly ordered group. Let Ω\Omega be the set of Archimedean classes of the nonzero elements of GG. Order Ω\Omega so that [a]≺[b][a] \prec [b] when aa is infinitely larger than bb. Let HH be the set of maps f ⁣:Ω→ℝf \colon \Omega \to \R whose support {ω:f(ω)≠0}\set{\omega : f(\omega) \ne 0} is well-ordered by ≺\prec. Make HH a group by pointwise addition, and set f>0f > 0 when f(ω0)>0f(\omega_0) > 0 for the least element ω0\omega_0 of the support of ff. Then HH is a linearly ordered abelian group, and there is an injective group homomorphism φ ⁣:G→H\varphi \colon G \to H with a≤ba \le b if and only if φ(a)≤φ(b)\varphi(a) \le \varphi(b).

Remarks

Hahn proved the theorem in 1907. For a nonzero a∈Ga \in G, let CaC_a be the smallest convex subgroup containing aa and DaD_a the largest convex subgroup not containing aa. The quotient Ca/DaC_a / D_a depends only on the class of aa. This quotient is Archimedean, so Hölder's theorem embeds it in the reals. The proofs then glue these embeddings together, using divisible hulls and Zorn's lemma. Example: ℤ2\Z^2 with the lexicographic order has two nonzero classes, and its image in H≅ℝ2H \cong \R^2 is the obvious one with the lexicographic order. When GG is Archimedean, Ω\Omega has one element and the theorem reduces to Hölder's theorem.

Going deeper

For a student

The theorem says that the groups HH, called Hahn groups, are universal for linearly ordered abelian groups with a given set of Archimedean classes. It is the group counterpart of the fact that the Hahn series fields are large enough to contain many valued fields.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionElement, definition, no evidence levelElementdefinitionGroup, concept, no evidence levelGroupconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectCommutativity, definition, no evidence levelCommutativitydefinitionSet, definition, no evidence levelSetdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionLeast element, definition, no evidence levelLeast elementdefinitionLexicographic order, definition, no evidence levelLexicographic orderdefinitionTotal order, definition, no evidence levelTotal orderdefinitionOrdered group, definition, no evidence levelOrdered groupdefinitionWell-order, definition, no evidence levelWell-orderdefinitionLinearly ordered group, definition, no evidence levelLinearly ordered groupdefinitionArchimedean equivalence classes, definition, no evidence levelArchimedean equivalen…definitionHahn embedding theorem, statement, no evidence levelHahn embedding theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary operation definition
  2. Element definition
  3. Group concept
  4. Proposition concept
  5. Real numbers object
  6. Commutativity definition
  7. Set definition
  8. Group homomorphism definition
  9. Implication definition
  10. Predicate definition
  11. Abelian group definition
  12. Ordered pair definition
  13. Universal quantifier definition
  14. Cartesian product definition
  15. Subset definition
  16. Binary relation definition
  17. Antisymmetric relation definition
  18. Reflexive relation definition
  19. Transitive relation definition
  20. Preorder definition
  21. Partial order definition
  22. Least element definition
  23. Lexicographic order definition
  24. Total order definition
  25. Ordered group definition
  26. Well-order definition
  27. Linearly ordered group definition
  28. Archimedean equivalence classes definition
  29. Hahn embedding theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Hahn_embedding_theorem
About this page
TagFVPXXFNVQY
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Kindstatement, theorem
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AboutLinearly ordered group, Archimedean equivalence classes
UsesLinearly ordered group, Archimedean equivalence classes, Abelian group, Well-order, Group homomorphism, Real numbers
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Changed2 November 2026

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