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())