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

""" 

Examples of finite monoids 

""" 

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

# 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.sets.family import Family 

from sage.structure.parent import Parent 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.structure.element_wrapper import ElementWrapper 

from sage.categories.all import Monoids 

from sage.rings.integer import Integer 

from sage.rings.integer_ring import ZZ 

class IntegerModMonoid(UniqueRepresentation, Parent): 

r""" 

An example of a finite monoid: the integers mod `n` 

This class illustrates a minimal implementation of a finite monoid. 

EXAMPLES:: 

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

An example of a finite multiplicative monoid: the integers modulo 12 

sage: S.category() 

Category of finitely generated finite enumerated monoids 

We conclude by running systematic tests on this monoid:: 

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

running ._test_an_element() . . . pass 

running ._test_associativity() . . . 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_enumerated_set_contains() . . . pass 

running ._test_enumerated_set_iter_cardinality() . . . pass 

running ._test_enumerated_set_iter_list() . . . pass 

running ._test_eq() . . . pass 

running ._test_new() . . . pass 

running ._test_not_implemented_methods() . . . pass 

running ._test_one() . . . pass 

running ._test_pickling() . . . pass 

running ._test_prod() . . . pass 

running ._test_some_elements() . . . pass 

""" 

def __init__(self, n = 12): 

r""" 

EXAMPLES:: 

sage: M = FiniteMonoids().example(6); M 

An example of a finite multiplicative monoid: the integers modulo 6 

TESTS:: 

sage: TestSuite(M).run() 

""" 

self.n = n 

Parent.__init__(self, category=Monoids().Finite().FinitelyGenerated()) 

def _repr_(self): 

r""" 

TESTS:: 

sage: M = FiniteMonoids().example() 

sage: M._repr_() 

'An example of a finite multiplicative monoid: the integers modulo 12' 

""" 

return "An example of a finite multiplicative monoid: the integers modulo %s"%self.n 

def semigroup_generators(self): 

r""" 

Returns a set of generators for ``self``, as per 

:meth:`Semigroups.ParentMethods.semigroup_generators`. 

Currently this returns all integers mod `n`, which is of 

course far from optimal! 

EXAMPLES:: 

sage: M = FiniteMonoids().example() 

sage: M.semigroup_generators() 

Family (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) 

""" 

return Family(tuple(self(ZZ(i)) for i in range(self.n))) 

@cached_method 

def one(self): 

r""" 

Return the one of the monoid, as per :meth:`Monoids.ParentMethods.one`. 

EXAMPLES:: 

sage: M = FiniteMonoids().example() 

sage: M.one() 

1 

""" 

return self(ZZ.one()) 

def product(self, x, y): 

r""" 

Return the product of two elements `x` and `y` of the monoid, as 

per :meth:`Semigroups.ParentMethods.product`. 

EXAMPLES:: 

sage: M = FiniteMonoids().example() 

sage: M.product(M(3), M(5)) 

3 

""" 

return self((x.value * y.value) % self.n) 

def an_element(self): 

r""" 

Returns an element of the monoid, as per :meth:`Sets.ParentMethods.an_element`. 

EXAMPLES:: 

sage: M = FiniteMonoids().example() 

sage: M.an_element() 

6 

""" 

return self(ZZ(42) % self.n) 

class Element (ElementWrapper): 

wrapped_class = Integer 

Example = IntegerModMonoid