Examples of commutative additive semigroups

sage.categories.examples.commutative_additive_semigroups.Example

alias of FreeCommutativeAdditiveSemigroup

class sage.categories.examples.commutative_additive_semigroups.FreeCommutativeAdditiveSemigroup(alphabet=('a', 'b', 'c', 'd'))

Bases: UniqueRepresentation, Parent

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 semigroup: the free commutative semigroup 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_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_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

class Element(parent, iterable)

Bases: ElementWrapper

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 semigroup: the free commutative semigroup 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}

additive_semigroup_generators()

Returns the generators of the semigroup.

EXAMPLES:

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

an_element()

Returns an element of the semigroup.

EXAMPLES:

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

summation(x, y)

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

EXAMPLES:

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