Coverage for local/lib/python2.7/site-packages/sage/categories/distributive_magmas_and_additive_magmas.py : 33%

r""" 

Distributive Magmas and Additive Magmas 

""" 

#***************************************************************************** 

# Copyright (C) 2010 Nicolas Borie <nicolas.borie@math.u-psud.fr> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#****************************************************************************** 

from sage.misc.lazy_import import LazyImport 

from sage.categories.category_with_axiom import CategoryWithAxiom 

from sage.categories.cartesian_product import CartesianProductsCategory 

class DistributiveMagmasAndAdditiveMagmas(CategoryWithAxiom): 

""" 

The category of sets `(S,+,*)` with `*` distributing on `+`. 

This is similar to a ring, but `+` and `*` are only required to be 

(additive) magmas. 

EXAMPLES:: 

sage: from sage.categories.distributive_magmas_and_additive_magmas import DistributiveMagmasAndAdditiveMagmas 

sage: C = DistributiveMagmasAndAdditiveMagmas(); C 

Category of distributive magmas and additive magmas 

sage: C.super_categories() 

[Category of magmas and additive magmas] 

TESTS:: 

sage: from sage.categories.magmas_and_additive_magmas import MagmasAndAdditiveMagmas 

sage: C is MagmasAndAdditiveMagmas().Distributive() 

True 

sage: C is (Magmas() & AdditiveMagmas()).Distributive() 

True 

sage: TestSuite(C).run() 

""" 

class AdditiveAssociative(CategoryWithAxiom): 

class AdditiveCommutative(CategoryWithAxiom): 

class AdditiveUnital(CategoryWithAxiom): 

class Associative(CategoryWithAxiom): 

AdditiveInverse = LazyImport('sage.categories.rngs', 'Rngs', at_startup=True) 

Unital = LazyImport('sage.categories.semirings', 'Semirings', at_startup=True) 

class ParentMethods: 

def _test_distributivity(self, **options): 

r""" 

Test the distributivity of `*` on `+` on (not necessarily 

all) elements of this set. 

INPUT: 

- ``options`` -- any keyword arguments accepted by :meth:`_tester` 

EXAMPLES: 

By default, this method runs the tests only on the 

elements returned by ``self.some_elements()``:: 

sage: NN.some_elements() 

[0, 1, 3, 42] 

sage: NN._test_distributivity() 

However, the elements tested can be customized with the 

``elements`` keyword argument:: 

sage: CC._test_distributivity(elements=[CC(0),CC(1),CC(3),CC(I)]) 

See the documentation for :class:`TestSuite` for more information. 

""" 

tester = self._tester(**options) 

S = tester.some_elements() 

from sage.misc.misc import some_tuples 

for x,y,z in some_tuples(tester.some_elements(), 3, tester._max_runs): 

# left distributivity 

tester.assertTrue(x * (y + z) == (x * y) + (x * z)) 

# right distributivity 

tester.assertTrue((x + y) * z == (x * z) + (y * z)) 

class CartesianProducts(CartesianProductsCategory): 

def extra_super_categories(self): 

""" 

Implement the fact that a Cartesian product of magmas distributing 

over additive magmas is a magma distributing over an 

additive magma. 

EXAMPLES:: 

sage: C = (Magmas() & AdditiveMagmas()).Distributive().CartesianProducts() 

sage: C.extra_super_categories(); 

[Category of distributive magmas and additive magmas] 

sage: C.axioms() 

frozenset({'Distributive'}) 

""" 

return [DistributiveMagmasAndAdditiveMagmas()]