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

r""" 

Example of a set with grading 

""" 

from sage.structure.parent import Parent 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.categories.sets_with_grading import SetsWithGrading 

from sage.rings.integer_ring import IntegerRing 

from sage.sets.finite_enumerated_set import FiniteEnumeratedSet 

class NonNegativeIntegers(UniqueRepresentation, Parent): 

r""" 

Non negative integers graded by themselves. 

EXAMPLES:: 

sage: E = SetsWithGrading().example() 

sage: E 

Non negative integers 

sage: E.graded_component(0) 

{0} 

sage: E.graded_component(100) 

{100} 

""" 

def __init__(self): 

r""" 

TESTS:: 

sage: TestSuite(SetsWithGrading().example()).run() 

""" 

Parent.__init__(self, category=SetsWithGrading(), facade=IntegerRing()) 

def an_element(self): 

r""" 

Returns 0. 

EXAMPLES:: 

sage: SetsWithGrading().example().an_element() 

0 

""" 

return 0 

def _repr_(self): 

r""" 

TESTS:: 

sage: SetsWithGrading().example() # indirect example 

Non negative integers 

""" 

return "Non negative integers" 

def graded_component(self, grade): 

r""" 

Returns the component with grade ``grade``. 

EXAMPLES:: 

sage: N = SetsWithGrading().example() 

sage: N.graded_component(65) 

{65} 

""" 

return FiniteEnumeratedSet([grade]) 

def grading(self, elt): 

r""" 

Returns the grade of ``elt``. 

EXAMPLES:: 

sage: N = SetsWithGrading().example() 

sage: N.grading(10) 

10 

""" 

return elt 

def generating_series(self, var='z'): 

r""" 

Returns `1 / (1-z)`. 

EXAMPLES:: 

sage: N = SetsWithGrading().example(); N 

Non negative integers 

sage: f = N.generating_series(); f 

1/(-z + 1) 

sage: LaurentSeriesRing(ZZ,'z')(f) 

1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + z^7 + z^8 + z^9 + z^10 + z^11 + z^12 + z^13 + z^14 + z^15 + z^16 + z^17 + z^18 + z^19 + O(z^20) 

""" 

from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing 

from sage.rings.integer import Integer 

R = PolynomialRing(IntegerRing(), var) 

z = R.gen() 

return Integer(1) / (Integer(1)-z) 

Example = NonNegativeIntegers