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# Graded algebras with basis

class `sage.categories.graded_algebras_with_basis.``GradedAlgebrasWithBasis`(base_category)

  Bases: `sage.categories.graded_modules.GradedModulesCategory`

  The category of graded algebras with a distinguished basis

  EXAMPLES:

      sage: C = GradedAlgebrasWithBasis(ZZ); C
      Category of graded algebras with basis over Integer Ring
      sage: sorted(C.super_categories(), key=str)
      [Category of algebras with basis over Integer Ring,
       Category of graded algebras over Integer Ring,
       Category of graded modules with basis over Integer Ring]

  TESTS:

      sage: TestSuite(C).run()

  class `ElementMethods`
    `degree`()

      The degree of this element.

      Note

      This raises an error if the element is not homogeneous. To obtain the maximum of the degrees of the homogeneous summands, use `maximal_degree()`

      EXAMPLES:

          sage: S = NonCommutativeSymmetricFunctions(QQ).S()
          sage: (x, y) = (S[2], S[3])
          sage: x.homogeneous_degree()
          2
          sage: (x^3 + 4*y^2).homogeneous_degree()
          6
          sage: ((1 + x)^3).homogeneous_degree()
          Traceback (most recent call last):
          ...
          ValueError: Element is not homogeneous.

      TESTS:

          sage: S = NonCommutativeSymmetricFunctions(QQ).S()
          sage: S.zero().degree()
          Traceback (most recent call last):
          ...
          ValueError: The zero element does not have a well-defined degree.

    `homogeneous_degree`()

      The degree of this element.

      Note

      This raises an error if the element is not homogeneous. To obtain the maximum of the degrees of the homogeneous summands, use `maximal_degree()`

      EXAMPLES:

          sage: S = NonCommutativeSymmetricFunctions(QQ).S()
          sage: (x, y) = (S[2], S[3])
          sage: x.homogeneous_degree()
          2
          sage: (x^3 + 4*y^2).homogeneous_degree()
          6
          sage: ((1 + x)^3).homogeneous_degree()
          Traceback (most recent call last):
          ...
          ValueError: Element is not homogeneous.

      TESTS:

          sage: S = NonCommutativeSymmetricFunctions(QQ).S()
          sage: S.zero().degree()
          Traceback (most recent call last):
          ...
          ValueError: The zero element does not have a well-defined degree.

    `is_homogeneous`()

      Return whether this element is homogeneous.

      EXAMPLES:

          sage: S = NonCommutativeSymmetricFunctions(QQ).S()
          sage: (x, y) = (S[2], S[3])
          sage: (3*x).is_homogeneous()
          True
          sage: (x^3 - y^2).is_homogeneous()
          True
          sage: ((x + y)^2).is_homogeneous()
          False

    `maximal_degree`()

      The maximum of the degrees of the homogeneous summands.

      EXAMPLES:

          sage: S = NonCommutativeSymmetricFunctions(QQ).S()
          sage: (x, y) = (S[2], S[3])
          sage: x.maximal_degree()
          2
          sage: (x^3 + 4*y^2).maximal_degree()
          6
          sage: ((1 + x)^3).maximal_degree()
          6

      TESTS:

          sage: S = NonCommutativeSymmetricFunctions(QQ).S()
          sage: S.zero().degree()
          Traceback (most recent call last):
          ...
          ValueError: The zero element does not have a well-defined degree.

  class `GradedAlgebrasWithBasis.``ParentMethods`

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