def __init__(self, category = None):
r"""
This quotient of the left zero semigroup of integers obtained by
setting `x=42` for any `x\geq 42`.
EXAMPLES::
sage: S = Semigroups().Subquotients().example(); S
An example of a (sub)quotient semigroup: a quotient of the left zero semigroup
sage: S.ambient()
An example of a semigroup: the left zero semigroup
sage: S(100)
42
sage: S(100) == S(42)
True
sage: S(1)*S(2) == S(1)
True
TESTS::
sage: TestSuite(S).run()
"""
if category is None:
category = Semigroups().Quotients()
Parent.__init__(self, category = category)
def ambient(self):
r"""
Returns the ambient semigroup.
EXAMPLES::
sage: S = Semigroups().Subquotients().example()
sage: S.ambient()
An example of a semigroup: the left zero semigroup
"""
return Semigroups().example()
def __init__(self, category=None):
r"""
An incompletely implemented subquotient semigroup, for testing purposes
EXAMPLES::
sage: S = sage.categories.examples.semigroups.IncompleteSubquotientSemigroup()
sage: S
A subquotient of An example of a semigroup: the left zero semigroup
TESTS::
sage: S._test_not_implemented_methods()
Traceback (most recent call last):
...
AssertionError: Not implemented method: lift
sage: TestSuite(S).run(verbose = True)
running ._test_an_element() . . . pass
running ._test_associativity() . . . fail
Traceback (most recent call last):
...
NotImplementedError: <abstract method retract at ...>
------------------------------------------------------------
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() . . . fail
Traceback (most recent call last):
...
AssertionError: Not implemented method: lift
------------------------------------------------------------
running ._test_pickling() . . . pass
running ._test_some_elements() . . . pass
The following tests failed: _test_associativity, _test_not_implemented_methods
"""
Parent.__init__(
self,
category=Semigroups().Subquotients().or_subcategory(category))
def __init__(self):
r"""
The left zero semigroup
EXAMPLES::
sage: S = Semigroups().example(); S
An example of a semigroup: the left zero semigroup
TESTS::
sage: TestSuite(S).run()
"""
Parent.__init__(self, category = Semigroups())
def __init__(self, alphabet=('a','b','c','d')):
r"""
The free semigroup.
INPUT::
- ``alphabet`` -- a tuple of strings: the generators of the semigroup
EXAMPLES::
sage: from sage.categories.examples.semigroups import FreeSemigroup
sage: F = FreeSemigroup(('a','b','c')); F
An example of a semigroup: the free semigroup generated by ('a', 'b', 'c')
TESTS::
sage: F == loads(dumps(F))
True
"""
self.alphabet = alphabet
Parent.__init__(self, category = Semigroups().FinitelyGenerated())