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

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

Python

>>> 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

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

Python

>>> 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

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

Python

>>> from sage.all import *
>>> (a,b,c,d) = S.additive_semigroup_generators()

We conclude by running systematic tests on this commutative monoid:

Sage

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

Python

>>> 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)

Bases: ElementWrapper

EXAMPLES:

Sage

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')

Python

>>> 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

sage: x.value
{'a': 2, 'b': 0, 'c': 1, 'd': 5}

Python

>>> from sage.all import *
>>> x.value
{'a': 2, 'b': 0, 'c': 1, 'd': 5}

additive_semigroup_generators()

Return the generators of the semigroup.

EXAMPLES:

Sage

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

Python

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

an_element()

Return an element of the semigroup.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> F = CommutativeAdditiveSemigroups().example()
>>> F.an_element()
a + 2*b + 3*c + 4*d

summation(x, y)

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

EXAMPLES:

Sage

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

Python

>>> 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