Examples of commutative additive monoids#

sage.categories.examples.commutative_additive_monoids.Example[source]#: alias of FreeCommutativeAdditiveMonoid

class sage.categories.examples.commutative_additive_monoids.FreeCommutativeAdditiveMonoid(alphabet=('a', 'b', 'c', 'd'))[source]#

Bases: FreeCommutativeAdditiveSemigroup

An example of a commutative additive monoid: the free commutative monoid

This class illustrates a minimal implementation of a commutative monoid.

EXAMPLES:

Sage

sage: S = CommutativeAdditiveMonoids().example(); S
An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd')

sage: S.category()
Category of commutative additive monoids

Python

>>> from sage.all import *
>>> S = CommutativeAdditiveMonoids().example(); S
An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd')

>>> S.category()
Category of commutative additive monoids

This is the free semigroup generated by:

Sage

sage: S.additive_semigroup_generators()
Family (a, b, c, d)

Python

>>> from sage.all import *
>>> S.additive_semigroup_generators()
Family (a, b, c, d)

with product rule given by \(a \times b = a\) for all \(a, b\):

Sage

sage: (a,b,c,d) = S.additive_semigroup_generators()

Python

>>> from sage.all import *
>>> (a,b,c,d) = S.additive_semigroup_generators()

We conclude by running systematic tests on this commutative monoid:

Sage

sage: TestSuite(S).run(verbose = True)
running ._test_additive_associativity() . . . pass
running ._test_an_element() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_construction() . . . 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_nonzero_equal() . . . 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_pickling() . . . pass
running ._test_some_elements() . . . pass
running ._test_zero() . . . pass

Python

>>> from sage.all import *
>>> TestSuite(S).run(verbose = True)
running ._test_additive_associativity() . . . pass
running ._test_an_element() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_construction() . . . 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_nonzero_equal() . . . 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_pickling() . . . pass
running ._test_some_elements() . . . pass
running ._test_zero() . . . pass

class Element(parent, iterable)[source]#: Bases: Element

zero()[source]#

Returns the zero of this additive monoid, as per CommutativeAdditiveMonoids.ParentMethods.zero().

EXAMPLES:

Sage

sage: M = CommutativeAdditiveMonoids().example(); M
An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd')
sage: M.zero()
0

Python

>>> from sage.all import *
>>> M = CommutativeAdditiveMonoids().example(); M
An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd')
>>> M.zero()
0