Examples of commutative additive semigroups#

sage.categories.examples.commutative_additive_semigroups.Example[source]#

alias of FreeCommutativeAdditiveSemigroup

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

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

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

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

This is the free semigroup generated by:

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

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

>>> from sage.all import *
>>> (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

>>> 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_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)[source]#

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')

>>> from sage.all import *
>>> F = CommutativeAdditiveSemigroups().example()
>>> x = F.element_class(F, (('a',Integer(4)), ('b', Integer(0)), ('a', Integer(2)), ('c', Integer(1)), ('d', Integer(5))))
>>> x
2*a + c + 5*d
>>> x.value
{'a': 2, 'b': 0, 'c': 1, 'd': 5}
>>> 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}

>>> from sage.all import *
>>> x.value
{'a': 2, 'b': 0, 'c': 1, 'd': 5}
additive_semigroup_generators()[source]#

Returns the generators of the semigroup.

EXAMPLES:

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

>>> from sage.all import *
>>> F = CommutativeAdditiveSemigroups().example()
>>> F.additive_semigroup_generators()
Family (a, b, c, d)
an_element()[source]#

Returns an element of the semigroup.

EXAMPLES:

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

>>> from sage.all import *
>>> F = CommutativeAdditiveSemigroups().example()
>>> F.an_element()
a + 2*b + 3*c + 4*d
summation(x, y)[source]#

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

>>> from sage.all import *
>>> F = CommutativeAdditiveSemigroups().example()
>>> (a,b,c,d) = F.additive_semigroup_generators()
>>> F.summation(a,b)
a + b
>>> (a+b) + (a+c)
2*a + b + c