Hahn series

E159923

Hahn series are formal power series with exponents in an ordered abelian group and well-ordered supports, providing a general framework for constructing large ordered fields that include structures like the surreal numbers.

All labels observed (3)

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Statements (47)

Predicate Object
instanceOf algebraic structure
formal power series
mathematical concept
ordered field construction
appearsIn construction of exponential fields
construction of real-closed fields
valuation theory of ordered fields
componentType coefficients from a base field or ring
exponents from an ordered abelian group
definedOver ordered abelian group
ensuresWellDefinedProductBy well-ordered support condition
fieldOfStudy algebra
model theory
non-Archimedean analysis
ordered algebra
valuation theory
generalizes Laurent series
Puiseux series
formal power series with integer exponents
hasAlternativeName Hahn series
surface form: Hahn generalized power series
hasCanonicalValuation support minimum exponent
hasConstraintOnSupport every nonempty subset of the support has a least element
hasExponentGroup ordered abelian group
hasMathematicianNamesake Hans Hahn
hasProperty can be equipped with a natural valuation
can be made into an ordered field
supports transfinite exponents
supports well-ordered sets of exponents
hasSupportCondition well-ordered support
isSpecialCaseOf generalized power series
namedAfter Hans Hahn
providesFrameworkFor constructing Hahn fields
embedding ordered fields
relatedConcept Levi-Civita field
Hahn series self-linksurface differs
surface form: Malcev–Neumann series

transseries
relatedTo surreal numbers
supportsOperation Cauchy-type product
termwise addition
typicalNotation K((G)) for coefficients in K and exponents in G
usedIn asymptotic differential algebra
construction of saturated valued fields
model-theoretic examples of non-Archimedean fields
study of ordered exponential fields
usedToConstruct large ordered fields
maximally valued fields
non-Archimedean ordered fields

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Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Surreal numbers relatedConcept Hahn series
Hahn series hasAlternativeName Hahn series
this entity surface form: Hahn generalized power series
Hahn series relatedConcept Hahn series self-linksurface differs
this entity surface form: Malcev–Neumann series