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r""" Additive semigroups """ #***************************************************************************** # Copyright (C) 2013 Nicolas M. Thiery <nthiery at users.sf.net> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #******************************************************************************
from sage.misc.cachefunc import cached_method from sage.misc.lazy_import import LazyImport from sage.categories.category_with_axiom import CategoryWithAxiom_singleton from sage.categories.cartesian_product import CartesianProductsCategory from sage.categories.algebra_functor import AlgebrasCategory from sage.categories.homsets import HomsetsCategory from sage.categories.additive_magmas import AdditiveMagmas
class AdditiveSemigroups(CategoryWithAxiom_singleton): """ The category of additive semigroups.
An *additive semigroup* is an associative :class:`additive magma <AdditiveMagmas>`, that is a set endowed with an operation `+` which is associative.
EXAMPLES::
sage: from sage.categories.additive_semigroups import AdditiveSemigroups sage: C = AdditiveSemigroups(); C Category of additive semigroups sage: C.super_categories() [Category of additive magmas] sage: C.all_super_categories() [Category of additive semigroups, Category of additive magmas, Category of sets, Category of sets with partial maps, Category of objects]
sage: C.axioms() frozenset({'AdditiveAssociative'}) sage: C is AdditiveMagmas().AdditiveAssociative() True
TESTS::
sage: TestSuite(C).run() """ _base_category_class_and_axiom = (AdditiveMagmas, "AdditiveAssociative")
AdditiveCommutative = LazyImport('sage.categories.commutative_additive_semigroups', 'CommutativeAdditiveSemigroups', at_startup=True) AdditiveUnital = LazyImport('sage.categories.additive_monoids', 'AdditiveMonoids', at_startup=True)
class ParentMethods: def _test_additive_associativity(self, **options): r""" Test associativity for (not necessarily all) elements of this additive semigroup.
INPUT:
- ``options`` -- any keyword arguments accepted by :meth:`_tester`
EXAMPLES:
By default, this method tests only the elements returned by ``self.some_elements()``::
sage: S = CommutativeAdditiveSemigroups().example() sage: S._test_additive_associativity()
However, the elements tested can be customized with the ``elements`` keyword argument::
sage: (a,b,c,d) = S.additive_semigroup_generators() sage: S._test_additive_associativity(elements = (a, b+c, d))
See the documentation for :class:`TestSuite` for more information. """
class Homsets(HomsetsCategory):
def extra_super_categories(self): r""" Implement the fact that a homset between two semigroups is a semigroup.
EXAMPLES::
sage: from sage.categories.additive_semigroups import AdditiveSemigroups sage: AdditiveSemigroups().Homsets().extra_super_categories() [Category of additive semigroups] sage: AdditiveSemigroups().Homsets().super_categories() [Category of homsets of additive magmas, Category of additive semigroups] """
class CartesianProducts(CartesianProductsCategory):
def extra_super_categories(self): """ Implement the fact that a Cartesian product of additive semigroups is an additive semigroup.
EXAMPLES::
sage: from sage.categories.additive_semigroups import AdditiveSemigroups sage: C = AdditiveSemigroups().CartesianProducts() sage: C.extra_super_categories() [Category of additive semigroups] sage: C.axioms() frozenset({'AdditiveAssociative'}) """
class Algebras(AlgebrasCategory):
def extra_super_categories(self): """ EXAMPLES::
sage: from sage.categories.additive_semigroups import AdditiveSemigroups sage: AdditiveSemigroups().Algebras(QQ).extra_super_categories() [Category of semigroups] sage: CommutativeAdditiveSemigroups().Algebras(QQ).super_categories() [Category of additive semigroup algebras over Rational Field, Category of additive commutative additive magma algebras over Rational Field] """
class ParentMethods:
@cached_method def algebra_generators(self): r""" Return the generators of this algebra, as per :meth:`MagmaticAlgebras.ParentMethods.algebra_generators() <.magmatic_algebras.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]} """
def product_on_basis(self, g1, g2): r""" Product, on basis elements, as per :meth:`MagmaticAlgebras.WithBasis.ParentMethods.product_on_basis() <sage.categories.magmatic_algebras.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] """
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