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refactor(RingTheory/HahnSeries) : several generalizations #12251

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@ScottCarnahan ScottCarnahan commented Apr 18, 2024

This PR is a container for several smaller PRs that refactor and generalize the existing Hahn series theory. It is (I think) all we need from Hahn series to get a palatable theory of vertex algebras off the ground (other prerequisites from Lie algebras will eventually come in a different PR). Major changes include:

  • equivalence between iterated Hahn series and Hahn series on Lex products.
  • introduce orderTop and leadingCoeff functions. Here, orderTop is a WithTop Γ-valued version of order that does not need [Zero Γ] and leadingCoeff returns the coefficient of the minimal element of support (or zero if empty).
  • introduce ordered and cancellative vector addition classes together with some basic theory e.g., finiteness of antidiagonals for partially well-ordered sets.
  • HahnSeries Γ R-module structure on HahnModule Γ' R V, when Γ is an OrderedCancelAddCommMonoid, Γ' is a PartialOrder with OrderedCancelVAdd Γ Γ', R is a semiring, and V is an R-module.
  • Move AddVal to a separate file - the underlying function is just orderTop, but the description of the valuation needs an additional import and an IsDomain hypothesis. Results that depended on AddVal are changed to use orderTop and generalized.

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@ScottCarnahan ScottCarnahan added t-algebra Algebra (groups, rings, fields, etc) t-order Order theory labels Apr 18, 2024
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@ScottCarnahan ScottCarnahan changed the title feat (RingTheory/HahnSeries) : refactor feat (RingTheory/HahnSeries) : generalize Apr 19, 2024
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@grunweg grunweg changed the title refactor (RingTheory/HahnSeries) : several generalizations refactor(RingTheory/HahnSeries) : several generalizations Oct 10, 2024
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