Examples of commutative additive semigroups¶
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sage.categories.examples.commutative_additive_semigroups.
Example
¶ An example of a commutative additive monoid: the free commutative monoid
This class illustrates a minimal implementation of a commutative additive monoid.
EXAMPLES:
sage: S = CommutativeAdditiveSemigroups().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 semigroups
This is the free semigroup generated by:
sage: S.additive_semigroup_generators() Family (a, b, c, d)
with product rule given by \(a \times b = a\) for all \(a, b\):
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_cardinality() . . . 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_new() . . . 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
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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 additive monoid.
EXAMPLES:
sage: S = CommutativeAdditiveSemigroups().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 semigroups
This is the free semigroup generated by:
sage: S.additive_semigroup_generators() Family (a, b, c, d)
with product rule given by \(a \times b = a\) for all \(a, b\):
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_cardinality() . . . 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_new() . . . 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