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

r""" 

Examples of 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.structure.element_wrapper import ElementWrapper 

from sage.categories.all import Monoids 

from .semigroups import FreeSemigroup 

from sage.sets.family import Family 

class FreeMonoid(FreeSemigroup): 

r""" 

An example of a monoid: the free monoid 

This class illustrates a minimal implementation of a monoid. For a 

full featured implementation of free monoids, see :func:`FreeMonoid`. 

EXAMPLES:: 

sage: S = Monoids().example(); S 

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

sage: S.category() 

Category of monoids 

This is the free semigroup generated by:: 

sage: S.semigroup_generators() 

Family ('a', 'b', 'c', 'd') 

with product rule given by concatenation of words:: 

sage: S('dab') * S('acb') 

'dabacb' 

and unit given by the empty word:: 

sage: S.one() 

'' 

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_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, alphabet=('a','b','c','d')): 

r""" 

The free monoid 

INPUT: 

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

EXAMPLES:: 

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

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

TESTS:: 

sage: TestSuite(M).run() 

""" 

self.alphabet = alphabet 

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

def _repr_(self): 

r""" 

TESTS:: 

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

sage: M._repr_() 

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

""" 

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

@cached_method 

def one(self): 

r""" 

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

EXAMPLES:: 

sage: M = Monoids().example(); M 

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

sage: M.one() 

'' 

""" 

return self("") 

@cached_method 

def monoid_generators(self): 

r""" 

Return the generators of this monoid. 

EXAMPLES:: 

sage: M = Monoids().example(); M 

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

sage: M.monoid_generators() 

Finite family {'a': 'a', 'c': 'c', 'b': 'b', 'd': 'd'} 

sage: a,b,c,d = M.monoid_generators() 

sage: a*d*c*b 

'adcb' 

""" 

return Family(self.alphabet, self) 

class Element (ElementWrapper): 

wrapped_class = str 

Example = FreeMonoid