Coverage for local/lib/python2.7/site-packages/sage/categories/additive_monoids.py : 14%

r""" 

Additive monoids 

""" 

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

# Copyright (C) 2013-2014 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.lazy_import import LazyImport 

from sage.categories.category_with_axiom import CategoryWithAxiom_singleton 

from sage.categories.additive_semigroups import AdditiveSemigroups 

from sage.categories.homsets import HomsetsCategory 

class AdditiveMonoids(CategoryWithAxiom_singleton): 

""" 

The category of additive monoids. 

An *additive monoid* is a unital :class:`additive semigroup 

<sage.categories.additive_semigroups.AdditiveSemigroups>`, that 

is a set endowed with a binary operation `+` which is associative 

and admits a zero (see :wikipedia:`Monoid`). 

EXAMPLES:: 

sage: from sage.categories.additive_monoids import AdditiveMonoids 

sage: C = AdditiveMonoids(); C 

Category of additive monoids 

sage: C.super_categories() 

[Category of additive unital additive magmas, Category of additive semigroups] 

sage: sorted(C.axioms()) 

['AdditiveAssociative', 'AdditiveUnital'] 

sage: from sage.categories.additive_semigroups import AdditiveSemigroups 

sage: C is AdditiveSemigroups().AdditiveUnital() 

True 

TESTS:: 

sage: C.Algebras(QQ).is_subcategory(AlgebrasWithBasis(QQ)) 

True 

sage: TestSuite(C).run() 

""" 

_base_category_class_and_axiom = (AdditiveSemigroups, "AdditiveUnital") 

AdditiveCommutative = LazyImport('sage.categories.commutative_additive_monoids', 'CommutativeAdditiveMonoids', at_startup=True) 

AdditiveInverse = LazyImport('sage.categories.additive_groups', 'AdditiveGroups', at_startup=True) 

class ParentMethods: 

def sum(self, args): 

r""" 

Return the sum of the elements in ``args``, as an element 

of ``self``. 

INPUT: 

- ``args`` -- a list (or iterable) of elements of ``self`` 

EXAMPLES:: 

sage: S = CommutativeAdditiveMonoids().example() 

sage: (a,b,c,d) = S.additive_semigroup_generators() 

sage: S.sum((a,b,a,c,a,b)) 

3*a + c + 2*b 

sage: S.sum(()) 

0 

sage: S.sum(()).parent() == S 

True 

""" 

return sum(args, self.zero()) 

class Homsets(HomsetsCategory): 

def extra_super_categories(self): 

""" 

Implement the fact that a homset between two monoids is 

associative. 

EXAMPLES:: 

sage: from sage.categories.additive_monoids import AdditiveMonoids 

sage: AdditiveMonoids().Homsets().extra_super_categories() 

[Category of additive semigroups] 

sage: AdditiveMonoids().Homsets().super_categories() 

[Category of homsets of additive unital additive magmas, Category of additive monoids] 

.. TODO:: 

This could be deduced from 

:meth:`AdditiveSemigroups.Homsets.extra_super_categories`. 

See comment in :meth:`Objects.SubcategoryMethods.Homsets`. 

""" 

return [AdditiveSemigroups()]