# Examples of commutative additive semigroups¶

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()


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_category() . . . pass
running ._test_elements() . . .
Running the test suite of self.an_element()
running ._test_category() . . . pass
running ._test_eq() . . . 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

class Element(parent, iterable)

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, 'b': 0, 'c': 1, '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, 'b': 0, 'c': 1, 'd': 5}


Returns the generators of the semigroup.

EXAMPLES:

sage: F = CommutativeAdditiveSemigroups().example()
Family (a, b, c, d)


Returns an element of the semigroup.

EXAMPLES:

sage: F = CommutativeAdditiveSemigroups().example()
sage: F.an_element()
a + 3*c + 2*b + 4*d


Returns the product of x and y in the semigroup, as per CommutativeAdditiveSemigroups.ParentMethods.summation().

EXAMPLES:

sage: F = CommutativeAdditiveSemigroups().example()
sage: F.summation(a,b)
a + b
sage: (a+b) + (a+c)
2*a + c + b