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.

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 for all :

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

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() . . . 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(value, parent)

Bases: sage.structure.element_wrapper.ElementWrapper

FreeMonoid.one(*args, **kwds)

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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