Coverage for local/lib/python2.7/site-packages/sage/categories/examples/commutative_additive_monoids.py : 100%

""" 

Examples of commutative additive monoids 

""" 

from __future__ import absolute_import 

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

# Copyright (C) 2008-2009 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.structure.parent import Parent 

from sage.categories.all import CommutativeAdditiveMonoids 

from .commutative_additive_semigroups import FreeCommutativeAdditiveSemigroup 

class FreeCommutativeAdditiveMonoid(FreeCommutativeAdditiveSemigroup): 

r""" 

An example of a commutative additive monoid: the free commutative monoid 

This class illustrates a minimal implementation of a commutative monoid. 

EXAMPLES:: 

sage: S = CommutativeAdditiveMonoids().example(); S 

An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd') 

sage: S.category() 

Category of commutative additive monoids 

This is the free semigroup generated by:: 

sage: S.additive_semigroup_generators() 

Family (a, b, c, d) 

with product rule given by $a \times b = a$ for all $a, b$:: 

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

We conclude by running systematic tests on this commutative monoid:: 

sage: TestSuite(S).run(verbose = True) 

running ._test_additive_associativity() . . . pass 

running ._test_an_element() . . . pass 

running ._test_cardinality() . . . pass 

running ._test_category() . . . pass 

running ._test_elements() . . . 

Running the test suite of self.an_element() 

running ._test_category() . . . pass 

running ._test_eq() . . . pass 

running ._test_new() . . . pass 

running ._test_nonzero_equal() . . . pass 

running ._test_not_implemented_methods() . . . pass 

running ._test_pickling() . . . pass 

pass 

running ._test_elements_eq_reflexive() . . . pass 

running ._test_elements_eq_symmetric() . . . pass 

running ._test_elements_eq_transitive() . . . pass 

running ._test_elements_neq() . . . pass 

running ._test_eq() . . . pass 

running ._test_new() . . . pass 

running ._test_not_implemented_methods() . . . pass 

running ._test_pickling() . . . pass 

running ._test_some_elements() . . . pass 

running ._test_zero() . . . pass 

""" 

def __init__(self, alphabet=('a','b','c','d')): 

r""" 

The free commutative monoid 

INPUT: 

- ``alphabet`` -- a tuple of strings: the generators of the monoid 

EXAMPLES:: 

sage: M = CommutativeAdditiveMonoids().example(alphabet=('a','b','c')); M 

An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c') 

TESTS:: 

sage: TestSuite(M).run() 

""" 

self.alphabet = alphabet 

Parent.__init__(self, category = CommutativeAdditiveMonoids()) 

def _repr_(self): 

r""" 

TESTS:: 

sage: M = CommutativeAdditiveMonoids().example(alphabet=('a','b','c')) 

sage: M._repr_() 

"An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c')" 

""" 

return "An example of a commutative monoid: the free commutative monoid generated by %s"%(self.alphabet,) 

@cached_method 

def zero(self): 

r""" 

Returns the zero of this additive monoid, as per :meth:`CommutativeAdditiveMonoids.ParentMethods.zero`. 

EXAMPLES:: 

sage: M = CommutativeAdditiveMonoids().example(); M 

An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd') 

sage: M.zero() 

0 

""" 

return self(()) 

class Element(FreeCommutativeAdditiveSemigroup.Element): 

def __bool__(self): 

""" 

Check if ``self`` is not the zero of the monoid 

EXAMPLES:: 

sage: M = CommutativeAdditiveMonoids().example() 

sage: bool(M.zero()) 

False 

sage: [bool(m) for m in M.additive_semigroup_generators()] 

[True, True, True, True] 

""" 

return any(x for x in self.value.values()) 

__nonzero__ = __bool__ 

Example = FreeCommutativeAdditiveMonoid