====== BigObject ValuatedMatroid<Addition, Scalar> ======
//from application [[..:matroid|matroid]]//\\
\\
 A valuated matroid. It is given by a matroid and some form of valuation, either on bases or circuits. It has two template parameters:

  ? Type Parameters:
  :: ''Addition'': Either ''[[..:common#Min |Min]]'' or ''[[..:common#Max |Max]]'', has no default on purpose.
  :: ''Scalar'': An ordered group in which the valuation lives, ''[[..:common#Rational |Rational]]'' by default.
  ? derived from:
  : ''[[..:matroid:Matroid |Matroid]]''
===== Properties =====

==== Combinatorics ====
 These properties capture combinatorial information of the object.  Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.
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{{anchor:subdivision:}}
  ?  **''SUBDIVISION''**
  :: This is the matroid subdivision of ''[[..:matroid:Matroid#POLYTOPE |POLYTOPE]]'' according to the lifting defined by  ''[[..:matroid:ValuatedMatroid#VALUATION_ON_BASES |VALUATION_ON_BASES]]'' (or minus ''[[..:matroid:ValuatedMatroid#VALUATION_ON_BASES |VALUATION_ON_BASES]]'' in the case of max).
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


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==== Valuation ====
 properties related to the valuation of the matroid.
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{{anchor:positive_valuation:}}
  ?  **''POSITIVE_VALUATION''**
  :: Whether the valuated matroid is transversal
    ? Type:
    :''[[..:common#Bool |Bool]]''


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{{anchor:transversal_valuated_matroid:}}
  ?  **''TRANSVERSAL_VALUATED_MATROID''**
  :: Whether the valuated matroid is transversal
    ? Type:
    :''[[..:common#Bool |Bool]]''


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{{anchor:valuation_on_bases:}}
  ?  **''VALUATION_ON_BASES''**
  :: Defines a valuation on each basis. Entry number i is a valuation on  the i-th element of ''[[..:matroid:Matroid#BASES |BASES]]''. Must fulfill the tropical Plücker relations.
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#TropicalNumber |TropicalNumber]]<Addition,Scalar%%>>%%''


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{{anchor:valuation_on_circuits:}}
  ?  **''VALUATION_ON_CIRCUITS''**
  :: Defines a valuation on each circuit. Row i is a representative of  the i-th element of ''[[..:matroid:Matroid#CIRCUITS |CIRCUITS]]''. Must fulfill the tropical circuit valuation axioms. The representative is normalized such that the first non-tropical-zero entry is 0.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<[[..:common#TropicalNumber |TropicalNumber]]<Addition,Scalar>,[[..:common#NonSymmetric |NonSymmetric]]>''


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