Example of a set with grading

sage.categories.examples.sets_with_grading.Example

alias of NonNegativeIntegers

class sage.categories.examples.sets_with_grading.NonNegativeIntegers

Bases: UniqueRepresentation, Parent

Non negative integers graded by themselves.

EXAMPLES:

sage: E = SetsWithGrading().example(); E
Non negative integers
sage: E in Sets().Infinite()
True
sage: E.graded_component(0)
{0}
sage: E.graded_component(100)
{100}

an_element()

Return 0.

EXAMPLES:

sage: SetsWithGrading().example().an_element()
0

generating_series(var='z')

Return \(1 / (1-z)\).

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)

graded_component(grade)

Return the component with grade grade.

EXAMPLES:

sage: N = SetsWithGrading().example()
sage: N.graded_component(65)
{65}

grading(elt)

Return the grade of elt.

EXAMPLES:

sage: N = SetsWithGrading().example()
sage: N.grading(10)
10