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

""" 

Examples of commutative additive semigroups 

""" 

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

# 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.structure.element_wrapper import ElementWrapper 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.categories.all import CommutativeAdditiveSemigroups 

from sage.sets.family import Family 

class FreeCommutativeAdditiveSemigroup(UniqueRepresentation, Parent): 

r""" 

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

This class illustrates a minimal implementation of a commutative additive monoid. 

EXAMPLES:: 

sage: S = CommutativeAdditiveSemigroups().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 semigroups 

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_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 

""" 

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

r""" 

The free commutative monoid 

INPUT: 

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

EXAMPLES:: 

sage: M = CommutativeAdditiveSemigroups().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 = CommutativeAdditiveSemigroups()) 

def _repr_(self): 

r""" 

EXAMPLES:: 

sage: M = CommutativeAdditiveSemigroups().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,) 

def summation(self, x, y): 

r""" 

Returns the product of ``x`` and ``y`` in the semigroup, as per 

:meth:`CommutativeAdditiveSemigroups.ParentMethods.summation`. 

EXAMPLES:: 

sage: F = CommutativeAdditiveSemigroups().example() 

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

sage: F.summation(a,b) 

a + b 

sage: (a+b) + (a+c) 

2*a + c + b 

""" 

assert x in self 

assert y in self 

return self((a, x.value[a] + y.value[a]) for a in self.alphabet) 

@cached_method 

def additive_semigroup_generators(self): 

r""" 

Returns the generators of the semigroup. 

EXAMPLES:: 

sage: F = CommutativeAdditiveSemigroups().example() 

sage: F.additive_semigroup_generators() 

Family (a, b, c, d) 

""" 

return Family( [self(((a,1),)) for a in self.alphabet] ) 

# FIXME: use this once the keys argument of FiniteFamily will be honoured 

# for the specifying the order of the elements in the family 

# return Family(self.alphabet, lambda a: self(((a,1),))) 

def an_element(self): 

r""" 

Returns an element of the semigroup. 

EXAMPLES:: 

sage: F = CommutativeAdditiveSemigroups().example() 

sage: F.an_element() 

a + 3*c + 2*b + 4*d 

""" 

return self((a, (ord(a)-96)) for a in self.alphabet) 

class Element(ElementWrapper): 

def __init__(self, parent, iterable): 

""" 

EXAMPLES:: 

sage: F = CommutativeAdditiveSemigroups().example() 

sage: x = F.element_class(F, (('a',4), ('b', 0), ('a', 2), ('c', 1), ('d', 5))) 

sage: x 

2*a + c + 5*d 

sage: x.value 

{'a': 2, 'b': 0, 'c': 1, 'd': 5} 

sage: x.parent() 

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

Internally, elements are represented as dense dictionaries which 

associate to each generator of the monoid its multiplicity. In 

order to get an element, we wrap the dictionary into an element 

via :class:`ElementWrapper`:: 

sage: x.value 

{'a': 2, 'b': 0, 'c': 1, 'd': 5} 

""" 

d = dict( (a,0) for a in parent.alphabet ) 

for (a, c) in iterable: 

d[a] = c 

ElementWrapper.__init__(self, parent, d) 

def _repr_(self): 

""" 

EXAMPLES:: 

sage: F = CommutativeAdditiveSemigroups().example() 

sage: F.an_element() # indirect doctest 

a + 3*c + 2*b + 4*d 

sage: F(()) 

0 

""" 

d = self.value 

result = ' + '.join( ("%s*%s"%(d[a],a) if d[a] != 1 else a) for a in d.keys() if d[a] != 0) 

return '0' if result == '' else result 

def __hash__(self): 

""" 

EXAMPLES:: 

sage: F = CommutativeAdditiveSemigroups().example() 

sage: type(hash(F.an_element())) 

<... 'int'> 

""" 

return hash(tuple(self.value.items())) 

Example = FreeCommutativeAdditiveSemigroup