Non Negative Integer Semiring#

class sage.rings.semirings.non_negative_integer_semiring.NonNegativeIntegerSemiring#

Bases: NonNegativeIntegers

A class for the semiring of the non negative integers

This parent inherits from the infinite enumerated set of non negative integers and endows it with its natural semiring structure.


sage: NonNegativeIntegerSemiring()
Non negative integer semiring

For convenience, NN is a shortcut for NonNegativeIntegerSemiring():

sage: NN == NonNegativeIntegerSemiring()

sage: NN.category()
Category of facade infinite enumerated commutative semirings

Here is a piece of the Cayley graph for the multiplicative structure:

sage: G = NN.cayley_graph(elements=range(9), generators=[0,1,2,3,5,7])          # needs sage.graphs
sage: G                                                                         # needs sage.graphs
Looped multi-digraph on 9 vertices
sage: G.plot()                                                                  # needs sage.graphs sage.plot
Graphics object consisting of 48 graphics primitives

This is the Hasse diagram of the divisibility order on NN.

sage: Poset(NN.cayley_graph(elements=[1..12], generators=[2,3,5,7,11])).show() # needs sage.combinat sage.graphs sage.plot

Note: as for NonNegativeIntegers, NN is currently just a “facade” parent; namely its elements are plain Sage Integers with Integer Ring as parent:

sage: x = NN(15); type(x)
<class 'sage.rings.integer.Integer'>
sage: x.parent()
Integer Ring
sage: x+3

Returns the additive semigroup generators of self.


sage: NN.additive_semigroup_generators()
Family (0, 1)