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# Additive Magmas

class `sage.categories.additive_magmas.``AdditiveMagmas`(s=None)

  Bases: `sage.categories.category_singleton.Category_singleton`

  The category of additive magmas.

  An additive magma is a set endowed with a binary operation \(+\).

  EXAMPLES:

      sage: AdditiveMagmas()
      Category of additive magmas
      sage: AdditiveMagmas().super_categories()
      [Category of sets]
      sage: AdditiveMagmas().all_super_categories()
      [Category of additive magmas, Category of sets, Category of sets with partial maps, Category of objects]

  The following axioms are defined by this category:

      sage: AdditiveMagmas().AdditiveAssociative()
      Category of additive semigroups
      sage: AdditiveMagmas().AdditiveUnital()
      Category of additive unital additive magmas
      sage: AdditiveMagmas().AdditiveCommutative()
      Category of additive commutative additive magmas
      sage: AdditiveMagmas().AdditiveUnital().AdditiveInverse()
      Category of additive inverse additive unital additive magmas
      sage: AdditiveMagmas().AdditiveAssociative().AdditiveCommutative()
      Category of commutative additive semigroups
      sage: AdditiveMagmas().AdditiveAssociative().AdditiveCommutative().AdditiveUnital()
      Category of commutative additive monoids
      sage: AdditiveMagmas().AdditiveAssociative().AdditiveCommutative().AdditiveUnital().AdditiveInverse()
      Category of commutative additive groups

  TESTS:

      sage: C = AdditiveMagmas()
      sage: TestSuite(C).run()

  `AdditiveAssociative`

    alias of `AdditiveSemigroups`

  class `AdditiveCommutative`(base_category)

    Bases: `sage.categories.category_with_axiom.CategoryWithAxiom_singleton`

    TESTS:

        sage: C = Sets.Finite(); C
        Category of finite sets
        sage: type(C)
        <class 'sage.categories.finite_sets.FiniteSets_with_category'>
        sage: type(C).__base__.__base__
        <class 'sage.categories.category_with_axiom.CategoryWithAxiom_singleton'>

        sage: TestSuite(C).run()

    class `Algebras`(category, *args)

      Bases: `sage.categories.algebra_functor.AlgebrasCategory`

      TESTS:

          sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
          sage: class FooBars(CovariantConstructionCategory):
          ...       _functor_category = "FooBars"
          sage: Category.FooBars = lambda self: FooBars.category_of(self)
          sage: C = FooBars(ModulesWithBasis(ZZ))
          sage: C
          Category of foo bars of modules with basis over Integer Ring
          sage: C.base_category()
          Category of modules with basis over Integer Ring
          sage: latex(C)
          \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
          sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
          sage: TestSuite(C).run()

      `extra_super_categories`()

        Implement the fact that the algebra of a commutative additive magmas is commutative.

        EXAMPLES:

            sage: AdditiveMagmas().AdditiveCommutative().Algebras(QQ).extra_super_categories()
            [Category of commutative magmas]

            sage: AdditiveMagmas().AdditiveCommutative().Algebras(QQ).super_categories()
            [Category of additive magma algebras over Rational Field,
             Category of commutative magmas]

    class `AdditiveMagmas.AdditiveCommutative.``CartesianProducts`(category, *args)

      Bases: `sage.categories.cartesian_product.CartesianProductsCategory`

      TESTS:

          sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
          sage: class FooBars(CovariantConstructionCategory):
          ...       _functor_category = "FooBars"
          sage: Category.FooBars = lambda self: FooBars.category_of(self)
          sage: C = FooBars(ModulesWithBasis(ZZ))
          sage: C
          Category of foo bars of modules with basis over Integer Ring
          sage: C.base_category()
          Category of modules with basis over Integer Ring
          sage: latex(C)
          \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
          sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
          sage: TestSuite(C).run()

      `extra_super_categories`()

        Implement the fact that a cartesian product of commutative additive magmas is a commutative additive magma.

        EXAMPLES:

            sage: C = AdditiveMagmas().AdditiveCommutative().CartesianProducts()
            sage: C.extra_super_categories();
            [Category of additive commutative additive magmas]
            sage: C.axioms()
            frozenset({'AdditiveCommutative'})

  class `AdditiveMagmas.``AdditiveUnital`(base_category)

    Bases: `sage.categories.category_with_axiom.CategoryWithAxiom_singleton`

    TESTS:

        sage: C = Sets.Finite(); C
        Category of finite sets
        sage: type(C)
        <class 'sage.categories.finite_sets.FiniteSets_with_category'>
        sage: type(C).__base__.__base__
        <class 'sage.categories.category_with_axiom.CategoryWithAxiom_singleton'>

        sage: TestSuite(C).run()

    class `AdditiveInverse`(base_category)

      Bases: `sage.categories.category_with_axiom.CategoryWithAxiom_singleton`

      TESTS:

          sage: C = Sets.Finite(); C
          Category of finite sets
          sage: type(C)
          <class 'sage.categories.finite_sets.FiniteSets_with_category'>
          sage: type(C).__base__.__base__
          <class 'sage.categories.category_with_axiom.CategoryWithAxiom_singleton'>

          sage: TestSuite(C).run()

      class `CartesianProducts`(category, *args)

        Bases: `sage.categories.cartesian_product.CartesianProductsCategory`

        TESTS:

            sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
            sage: class FooBars(CovariantConstructionCategory):
            ...       _functor_category = "FooBars"
            sage: Category.FooBars = lambda self: FooBars.category_of(self)
            sage: C = FooBars(ModulesWithBasis(ZZ))
            sage: C
            Category of foo bars of modules with basis over Integer Ring
            sage: C.base_category()
            Category of modules with basis over Integer Ring
            sage: latex(C)
            \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
            sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
            sage: TestSuite(C).run()

        `extra_super_categories`()

          Implement the fact that a cartesian product of additive magmas with inverses is an additive magma with inverse.

          EXAMPLES:

              sage: C = AdditiveMagmas().AdditiveUnital().AdditiveInverse().CartesianProducts()
              sage: C.extra_super_categories();
              [Category of additive inverse additive unital additive magmas]
              sage: sorted(C.axioms())
              ['AdditiveInverse', 'AdditiveUnital']

    class `AdditiveMagmas.AdditiveUnital.``Algebras`(category, *args)

      Bases: `sage.categories.algebra_functor.AlgebrasCategory`

      TESTS:

          sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
          sage: class FooBars(CovariantConstructionCategory):
          ...       _functor_category = "FooBars"
          sage: Category.FooBars = lambda self: FooBars.category_of(self)
          sage: C = FooBars(ModulesWithBasis(ZZ))
          sage: C
          Category of foo bars of modules with basis over Integer Ring
          sage: C.base_category()
          Category of modules with basis over Integer Ring
          sage: latex(C)
          \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
          sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
          sage: TestSuite(C).run()

      class `ParentMethods`
        `one_basis`()

          Return the zero of this additive magma, which index the one of this algebra, as per `AlgebrasWithBasis.ParentMethods.one_basis()`.

          EXAMPLES:

              sage: S = CommutativeAdditiveMonoids().example(); S
              An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd')
              sage: A = S.algebra(ZZ)
              sage: A.one_basis()
              0
              sage: A.one()
              B[0]
              sage: A(3)
              3*B[0]

      `AdditiveMagmas.AdditiveUnital.Algebras.``extra_super_categories`()

        EXAMPLES:

            sage: C = AdditiveMagmas().AdditiveUnital().Algebras(QQ)
            sage: C.extra_super_categories()
            [Category of unital magmas]

            sage: C.super_categories()
            [Category of unital algebras with basis over Rational Field, Category of additive magma algebras over Rational Field]

    class `AdditiveMagmas.AdditiveUnital.``CartesianProducts`(category, *args)

      Bases: `sage.categories.cartesian_product.CartesianProductsCategory`

      TESTS:

          sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
          sage: class FooBars(CovariantConstructionCategory):
          ...       _functor_category = "FooBars"
          sage: Category.FooBars = lambda self: FooBars.category_of(self)
          sage: C = FooBars(ModulesWithBasis(ZZ))
          sage: C
          Category of foo bars of modules with basis over Integer Ring
          sage: C.base_category()
          Category of modules with basis over Integer Ring
          sage: latex(C)
          \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
          sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
          sage: TestSuite(C).run()

      class `ElementMethods`
      class `AdditiveMagmas.AdditiveUnital.CartesianProducts.``ParentMethods`
        `zero`()

          Returns the zero of this group

          EXAMPLE:

              sage: GF(8,'x').cartesian_product(GF(5)).zero()
              (0, 0)

      `AdditiveMagmas.AdditiveUnital.CartesianProducts.``extra_super_categories`()

        Implement the fact that a cartesian product of unital additive magmas is a unital additive magma.

        EXAMPLES:

            sage: C = AdditiveMagmas().AdditiveUnital().CartesianProducts()
            sage: C.extra_super_categories();
            [Category of additive unital additive magmas]
            sage: C.axioms()
            frozenset({'AdditiveUnital'})

    class `AdditiveMagmas.AdditiveUnital.``ElementMethods`
    class `AdditiveMagmas.AdditiveUnital.``Homsets`(category, *args)

      Bases: `sage.categories.homsets.HomsetsCategory`

      TESTS:

          sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
          sage: class FooBars(CovariantConstructionCategory):
          ...       _functor_category = "FooBars"
          sage: Category.FooBars = lambda self: FooBars.category_of(self)
          sage: C = FooBars(ModulesWithBasis(ZZ))
          sage: C
          Category of foo bars of modules with basis over Integer Ring
          sage: C.base_category()
          Category of modules with basis over Integer Ring
          sage: latex(C)
          \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
          sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
          sage: TestSuite(C).run()

      class `ParentMethods`
        `zero`()

          EXAMPLES:

              sage: R = QQ['x']
              sage: H = Hom(ZZ, R, AdditiveMagmas().AdditiveUnital())
              sage: f = H.zero()
              sage: f
              Generic morphism:
                From: Integer Ring
                To:   Univariate Polynomial Ring in x over Rational Field
              sage: f(3)
              0
              sage: f(3) is R.zero()
              True

          TESTS:

          > sage: TestSuite(f).run()

      `AdditiveMagmas.AdditiveUnital.Homsets.``extra_super_categories`()

        Implement the fact that a homset between two unital additive magmas is a unital additive magma.

        EXAMPLES:

            sage: AdditiveMagmas().AdditiveUnital().Homsets().extra_super_categories()
            [Category of additive unital additive magmas]
            sage: AdditiveMagmas().AdditiveUnital().Homsets().super_categories()
            [Category of additive unital additive magmas]

    class `AdditiveMagmas.AdditiveUnital.``ParentMethods`
      `zero`()

        Return the zero of this additive magma, that is the unique neutral element for \(+\).

        The default implementation is to coerce `0` into `self`.

        It is recommended to override this method because the coercion from the integers:

        - is not always meaningful (except for \(0\)), and
        - often uses `self.zero()` otherwise.

        EXAMPLES:

            sage: S = CommutativeAdditiveMonoids().example()
            sage: S.zero()
            0

      `zero_element`()

        Backward compatibility alias for `self.zero()`.

        TESTS:

            sage: S = CommutativeAdditiveMonoids().example()
            sage: S.zero_element()
            0

    class `AdditiveMagmas.AdditiveUnital.``SubcategoryMethods`
      `AdditiveInverse`()

        Return the full subcategory of the additive inverse objects of `self`.

        An inverse `additive magma` is a `unital additive magma` such that every element admits both an additive inverse on the left and on the right. Such an additive magma is also called an additive loop.

        See also

        Wikipedia article Inverse_element, Wikipedia article Quasigroup

        EXAMPLES:

            sage: AdditiveMagmas().AdditiveUnital().AdditiveInverse()
            Category of additive inverse additive unital additive magmas
            sage: from sage.categories.additive_monoids import AdditiveMonoids
            sage: AdditiveMonoids().AdditiveInverse()
            Category of additive groups

        TESTS:

            sage: TestSuite(AdditiveMagmas().AdditiveUnital().AdditiveInverse()).run()
            sage: CommutativeAdditiveMonoids().AdditiveInverse.__module__
            'sage.categories.additive_magmas'

    class `AdditiveMagmas.AdditiveUnital.``WithRealizations`(category, *args)

      Bases: `sage.categories.with_realizations.WithRealizationsCategory`

      TESTS:

          sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
          sage: class FooBars(CovariantConstructionCategory):
          ...       _functor_category = "FooBars"
          sage: Category.FooBars = lambda self: FooBars.category_of(self)
          sage: C = FooBars(ModulesWithBasis(ZZ))
          sage: C
          Category of foo bars of modules with basis over Integer Ring
          sage: C.base_category()
          Category of modules with basis over Integer Ring
          sage: latex(C)
          \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
          sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
          sage: TestSuite(C).run()

      class `ParentMethods`
        `zero`()

          Return the zero of this unital additive magma.

          This default implementation returns the zero of the realization of `self` given by `a_realization()`.

          EXAMPLES:

              sage: A = Sets().WithRealizations().example(); A
              The subset algebra of {1, 2, 3} over Rational Field
              sage: A.zero.__module__
              'sage.categories.additive_magmas'
              sage: A.zero()
              0

          TESTS:

              sage: A.zero() is A.a_realization().zero()
              True
              sage: A._test_zero()

    `AdditiveMagmas.AdditiveUnital.``additional_structure`()

      Return whether `self` is a structure category.

      See also

      `Category.additional_structure()`

      The category of unital additive magmas defines the zero as additional structure, and this zero shall be preserved by morphisms.

      EXAMPLES:

          sage: AdditiveMagmas().AdditiveUnital().additional_structure()
          Category of additive unital additive magmas

  class `AdditiveMagmas.``Algebras`(category, *args)

    Bases: `sage.categories.algebra_functor.AlgebrasCategory`

    TESTS:

        sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
        sage: class FooBars(CovariantConstructionCategory):
        ...       _functor_category = "FooBars"
        sage: Category.FooBars = lambda self: FooBars.category_of(self)
        sage: C = FooBars(ModulesWithBasis(ZZ))
        sage: C
        Category of foo bars of modules with basis over Integer Ring
        sage: C.base_category()
        Category of modules with basis over Integer Ring
        sage: latex(C)
        \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
        sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
        sage: TestSuite(C).run()

    class `ParentMethods`
      `algebra_generators`()

        The generators of this algebra, as per `MagmaticAlgebras.ParentMethods.algebra_generators()`.

        They correspond to the generators of the additive semigroup.

        EXAMPLES:

            sage: S = CommutativeAdditiveSemigroups().example(); S
            An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd')
            sage: A = S.algebra(QQ)
            sage: A.algebra_generators()
            Finite family {0: B[a], 1: B[b], 2: B[c], 3: B[d]}

      `product_on_basis`(g1, g2)

        Product, on basis elements, as per `MagmaticAlgebras.WithBasis.ParentMethods.product_on_basis()`.

        The product of two basis elements is induced by the addition of the corresponding elements of the group.

        EXAMPLES:

            sage: S = CommutativeAdditiveSemigroups().example(); S
            An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd')
            sage: A = S.algebra(QQ)
            sage: a,b,c,d = A.algebra_generators()
            sage: a * b + b * d * c
            B[c + b + d] + B[a + b]

    `AdditiveMagmas.Algebras.``extra_super_categories`()

      EXAMPLES:

          sage: AdditiveMagmas().Algebras(QQ).extra_super_categories()
          [Category of magmatic algebras with basis over Rational Field]

          sage: AdditiveMagmas().Algebras(QQ).super_categories()
          [Category of magmatic algebras with basis over Rational Field, Category of set algebras over Rational Field]

  class `AdditiveMagmas.``CartesianProducts`(category, *args)

    Bases: `sage.categories.cartesian_product.CartesianProductsCategory`

    TESTS:

        sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
        sage: class FooBars(CovariantConstructionCategory):
        ...       _functor_category = "FooBars"
        sage: Category.FooBars = lambda self: FooBars.category_of(self)
        sage: C = FooBars(ModulesWithBasis(ZZ))
        sage: C
        Category of foo bars of modules with basis over Integer Ring
        sage: C.base_category()
        Category of modules with basis over Integer Ring
        sage: latex(C)
        \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
        sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
        sage: TestSuite(C).run()

    class `ElementMethods`
    `AdditiveMagmas.CartesianProducts.``extra_super_categories`()

      Implement the fact that a cartesian product of additive magmas is an additive magma.

      EXAMPLES:

          sage: C = AdditiveMagmas().CartesianProducts()
          sage: C.extra_super_categories()
          [Category of additive magmas]
          sage: C.super_categories()
          [Category of additive magmas, Category of Cartesian products of sets]
          sage: C.axioms()
          frozenset()

  class `AdditiveMagmas.``ElementMethods`
  class `AdditiveMagmas.``Homsets`(category, *args)

    Bases: `sage.categories.homsets.HomsetsCategory`

    TESTS:

        sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
        sage: class FooBars(CovariantConstructionCategory):
        ...       _functor_category = "FooBars"
        sage: Category.FooBars = lambda self: FooBars.category_of(self)
        sage: C = FooBars(ModulesWithBasis(ZZ))
        sage: C
        Category of foo bars of modules with basis over Integer Ring
        sage: C.base_category()
        Category of modules with basis over Integer Ring
        sage: latex(C)
        \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
        sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
        sage: TestSuite(C).run()

    `extra_super_categories`()

      Implement the fact that a homset between two magmas is a magma.

      EXAMPLES:

          sage: AdditiveMagmas().Homsets().extra_super_categories()
          [Category of additive magmas]
          sage: AdditiveMagmas().Homsets().super_categories()
          [Category of additive magmas]

  class `AdditiveMagmas.``ParentMethods`
    `addition_table`(names='letters', elements=None)

      Return a table describing the addition operation.

      Note

      The order of the elements in the row and column headings is equal to the order given by the table’s `column_keys()` method. The association can also be retrieved with the `translation()` method.

      INPUT:

      - `names` – the type of names used:
        - `'letters'` - lowercase ASCII letters are used for a base 26 representation of the elements’ positions in the list given by `column_keys()`, padded to a common width with leading ‘a’s.
        - `'digits'` - base 10 representation of the elements’ positions in the list given by `column_keys()`, padded to a common width with leading zeros.
        - `'elements'` - the string representations of the elements themselves.
        - a list - a list of strings, where the length of the list equals the number of elements.
      - `elements` – (default: `None`) A list of elements of the additive magma, in forms that can be coerced into the structure, eg. their string representations. This may be used to impose an alternate ordering on the elements, perhaps when this is used in the context of a particular structure. The default is to use whatever ordering the `S.list` method returns. Or the `elements` can be a subset which is closed under the operation. In particular, this can be used when the base set is infinite.

      OUTPUT:

      The addition table as an object of the class `OperationTable` which defines several methods for manipulating and displaying the table. See the documentation there for full details to supplement the documentation here.

      EXAMPLES:

      All that is required is that an algebraic structure has an addition defined.The default is to represent elements as lowercase ASCII letters.

          sage: R=IntegerModRing(5)
          sage: R.addition_table()
          +  a b c d e
           +----------
          a| a b c d e
          b| b c d e a
          c| c d e a b
          d| d e a b c
          e| e a b c d

      The `names` argument allows displaying the elements in different ways. Requesting `elements` will use the representation of the elements of the set. Requesting `digits` will include leading zeros as padding.

          sage: R=IntegerModRing(11)
          sage: P=R.addition_table(names='elements')
          sage: P
           +   0  1  2  3  4  5  6  7  8  9 10
            +---------------------------------
           0|  0  1  2  3  4  5  6  7  8  9 10
           1|  1  2  3  4  5  6  7  8  9 10  0
           2|  2  3  4  5  6  7  8  9 10  0  1
           3|  3  4  5  6  7  8  9 10  0  1  2
           4|  4  5  6  7  8  9 10  0  1  2  3
           5|  5  6  7  8  9 10  0  1  2  3  4
           6|  6  7  8  9 10  0  1  2  3  4  5
           7|  7  8  9 10  0  1  2  3  4  5  6
           8|  8  9 10  0  1  2  3  4  5  6  7
           9|  9 10  0  1  2  3  4  5  6  7  8
          10| 10  0  1  2  3  4  5  6  7  8  9

          sage: T=R.addition_table(names='digits')
          sage: T
          +  00 01 02 03 04 05 06 07 08 09 10
            +---------------------------------
          00| 00 01 02 03 04 05 06 07 08 09 10
          01| 01 02 03 04 05 06 07 08 09 10 00
          02| 02 03 04 05 06 07 08 09 10 00 01
          03| 03 04 05 06 07 08 09 10 00 01 02
          04| 04 05 06 07 08 09 10 00 01 02 03
          05| 05 06 07 08 09 10 00 01 02 03 04
          06| 06 07 08 09 10 00 01 02 03 04 05
          07| 07 08 09 10 00 01 02 03 04 05 06
          08| 08 09 10 00 01 02 03 04 05 06 07
          09| 09 10 00 01 02 03 04 05 06 07 08
          10| 10 00 01 02 03 04 05 06 07 08 09

      Specifying the elements in an alternative order can provide more insight into how the operation behaves.

          sage: S=IntegerModRing(7)
          sage: elts = [0, 3, 6, 2, 5, 1, 4]
          sage: S.addition_table(elements=elts)
          +  a b c d e f g
           +--------------
          a| a b c d e f g
          b| b c d e f g a
          c| c d e f g a b
          d| d e f g a b c
          e| e f g a b c d
          f| f g a b c d e
          g| g a b c d e f

      The `elements` argument can be used to provide a subset of the elements of the structure. The subset must be closed under the operation. Elements need only be in a form that can be coerced into the set. The `names` argument can also be used to request that the elements be represented with their usual string representation.

          sage: T=IntegerModRing(12)
          sage: elts=[0, 3, 6, 9]
          sage: T.addition_table(names='elements', elements=elts)
          +  0 3 6 9
           +--------
          0| 0 3 6 9
          3| 3 6 9 0
          6| 6 9 0 3
          9| 9 0 3 6

      The table returned can be manipulated in various ways. See the documentation for `OperationTable` for more comprehensive documentation.

          sage: R=IntegerModRing(3)
          sage: T=R.addition_table()
          sage: T.column_keys()
          (0, 1, 2)
          sage: sorted(T.translation().items())
          [('a', 0), ('b', 1), ('c', 2)]
          sage: T.change_names(['x', 'y', 'z'])
          sage: sorted(T.translation().items())
          [('x', 0), ('y', 1), ('z', 2)]
          sage: T
          +  x y z
           +------
          x| x y z
          y| y z x
          z| z x y

    `summation`(x, y)

      Return the sum of `x` and `y`.

      The binary addition operator of this additive magma.

      INPUT:

      > - `x`, `y` – elements of this additive magma

      EXAMPLES:

          sage: S = CommutativeAdditiveSemigroups().example()
          sage: (a,b,c,d) = S.additive_semigroup_generators()
          sage: S.summation(a, b)
          a + b

      A parent in `AdditiveMagmas()` must either implement `summation()` in the parent class or `_add_` in the element class. By default, the addition method on elements `x._add_(y)` calls `S.summation(x,y)`, and reciprocally.

      As a bonus effect, `S.summation` by itself models the binary function from `S` to `S`:

          sage: bin = S.summation
          sage: bin(a,b)
          a + b

      Here, `S.summation` is just a bound method. Whenever possible, it is recommended to enrich `S.summation` with extra mathematical structure. Lazy attributes can come handy for this.

      Todo

      Add an example.

    `summation_from_element_class_add`(x, y)

      Return the sum of `x` and `y`.

      The binary addition operator of this additive magma.

      INPUT:

      > - `x`, `y` – elements of this additive magma

      EXAMPLES:

          sage: S = CommutativeAdditiveSemigroups().example()
          sage: (a,b,c,d) = S.additive_semigroup_generators()
          sage: S.summation(a, b)
          a + b

      A parent in `AdditiveMagmas()` must either implement `summation()` in the parent class or `_add_` in the element class. By default, the addition method on elements `x._add_(y)` calls `S.summation(x,y)`, and reciprocally.

      As a bonus effect, `S.summation` by itself models the binary function from `S` to `S`:

          sage: bin = S.summation
          sage: bin(a,b)
          a + b

      Here, `S.summation` is just a bound method. Whenever possible, it is recommended to enrich `S.summation` with extra mathematical structure. Lazy attributes can come handy for this.

      Todo

      Add an example.

  class `AdditiveMagmas.``SubcategoryMethods`
    `AdditiveAssociative`()

      Return the full subcategory of the additive associative objects of `self`.

      An `additive magma` \(M\) is associative if, for all \(x,y,z \in M\),

      \[x + (y + z) = (x + y) + z\]

      See also

      Wikipedia article Associative_property

      EXAMPLES:

          sage: AdditiveMagmas().AdditiveAssociative()
          Category of additive semigroups

      TESTS:

          sage: TestSuite(AdditiveMagmas().AdditiveAssociative()).run()
          sage: Rings().AdditiveAssociative.__module__
          'sage.categories.additive_magmas'

    `AdditiveCommutative`()

      Return the full subcategory of the commutative objects of `self`.

      An `additive magma` \(M\) is commutative if, for all \(x,y \in M\),

      \[x + y = y + x\]

      See also

      Wikipedia article Commutative_property

      EXAMPLES:

          sage: AdditiveMagmas().AdditiveCommutative()
          Category of additive commutative additive magmas
          sage: AdditiveMagmas().AdditiveAssociative().AdditiveUnital().AdditiveCommutative()
          Category of commutative additive monoids
          sage: _ is CommutativeAdditiveMonoids()
          True

      TESTS:

          sage: TestSuite(AdditiveMagmas().AdditiveCommutative()).run()
          sage: Rings().AdditiveCommutative.__module__
          'sage.categories.additive_magmas'

    `AdditiveUnital`()

      Return the subcategory of the unital objects of `self`.

      An `additive magma` \(M\) is unital if it admits an element \(0\), called neutral element, such that for all \(x \in M\),

      \[0 + x = x + 0 = x\]

      This element is necessarily unique, and should be provided as `M.zero()`.

      See also

      Wikipedia article Unital_magma#unital

      EXAMPLES:

          sage: AdditiveMagmas().AdditiveUnital()
          Category of additive unital additive magmas
          sage: from sage.categories.additive_semigroups import AdditiveSemigroups
          sage: AdditiveSemigroups().AdditiveUnital()
          Category of additive monoids
          sage: CommutativeAdditiveMonoids().AdditiveUnital()
          Category of commutative additive monoids

      TESTS:

          sage: TestSuite(AdditiveMagmas().AdditiveUnital()).run()
          sage: CommutativeAdditiveSemigroups().AdditiveUnital.__module__
          'sage.categories.additive_magmas'

  `AdditiveMagmas.``super_categories`()

    EXAMPLES:

        sage: AdditiveMagmas().super_categories()
        [Category of sets]

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