Examples of monoids

sage.categories.examples.monoids.Example

alias of FreeMonoid

class sage.categories.examples.monoids.FreeMonoid(alphabet=('a', 'b', 'c', 'd'))

Bases: sage.categories.examples.semigroups.FreeSemigroup

An example of a monoid: the free monoid

This class illustrates a minimal implementation of a monoid. For a full featured implementation of free monoids, see FreeMonoid().

EXAMPLES:

sage: S = Monoids().example(); S
An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')

sage: S.category()
Category of monoids

This is the free semigroup generated by:

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

with product rule given by concatenation of words:

sage: S('dab') * S('acb')
'dabacb'

and unit given by the empty word:

sage: S.one()
''

We conclude by running systematic tests on this monoid:

sage: TestSuite(S).run(verbose = True)
running ._test_an_element() . . . pass
running ._test_associativity() . . . 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_one() . . . pass
running ._test_pickling() . . . pass
running ._test_prod() . . . pass
running ._test_some_elements() . . . pass
class Element

Bases: sage.structure.element_wrapper.ElementWrapper

EXAMPLES:

sage: from sage.structure.element_wrapper import DummyParent
sage: a = ElementWrapper(DummyParent("A parent"), 1)

TESTS:

sage: TestSuite(a).run(skip = "_test_category")

sage: a = ElementWrapper(1, DummyParent("A parent"))
doctest:...: DeprecationWarning: the first argument must be a parent
See http://trac.sagemath.org/14519 for details.

Note

ElementWrapper is not intended to be used directly, hence the failing category test.

FreeMonoid.one()

Returns the one of the monoid, as per Monoids.ParentMethods.one().

EXAMPLES:

sage: M = Monoids().example(); M
An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')
sage: M.one()
''

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