alias of FreeCommutativeAdditiveMonoid
Bases: sage.categories.examples.commutative_additive_semigroups.FreeCommutativeAdditiveSemigroup
An example of a commutative additive monoid: the free commutative monoid
This class illustrates a minimal implementation of a commutative monoid.
EXAMPLES:
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
This is the free semigroup generated by:
sage: S.additive_semigroup_generators()
Family (a, b, c, d)
with product rule given by for all
:
sage: (a,b,c,d) = S.additive_semigroup_generators()
We conclude by running systematic tests on this commutative monoid:
sage: TestSuite(S).run(verbose = True)
running ._test_additive_associativity() . . . pass
running ._test_an_element() . . . 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_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_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
running ._test_some_elements() . . . pass
running ._test_zero() . . . pass
Bases: sage.categories.examples.commutative_additive_semigroups.FreeCommutativeAdditiveSemigroup.Element
EXAMPLES:
sage: F = CommutativeAdditiveSemigroups().example()
sage: x = F.element_class(F, (('a',4), ('b', 0), ('a', 2), ('c', 1), ('d', 5)))
sage: x
2*a + c + 5*d
sage: x.value
{'a': 2, 'c': 1, 'b': 0, 'd': 5}
sage: x.parent()
An example of a commutative monoid: the free commutative monoid generated by ('a', 'b', 'c', 'd')
Internally, elements are represented as dense dictionaries which associate to each generator of the monoid its multiplicity. In order to get an element, we wrap the dictionary into an element via ElementWrapper:
sage: x.value
{'a': 2, 'c': 1, 'b': 0, 'd': 5}
Returns the zero of this additive monoid, as per CommutativeAdditiveMonoids.ParentMethods.zero().
EXAMPLES:
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