Examples of finite semigroups#
- sage.categories.examples.finite_semigroups.Example[source]#
alias of
LeftRegularBand
- class sage.categories.examples.finite_semigroups.LeftRegularBand(alphabet=('a', 'b', 'c', 'd'))[source]#
Bases:
UniqueRepresentation
,Parent
An example of a finite semigroup
This class provides a minimal implementation of a finite semigroup.
EXAMPLES:
sage: S = FiniteSemigroups().example(); S An example of a finite semigroup: the left regular band generated by ('a', 'b', 'c', 'd')
>>> from sage.all import * >>> S = FiniteSemigroups().example(); S An example of a finite semigroup: the left regular band generated by ('a', 'b', 'c', 'd')
This is the semigroup generated by:
sage: S.semigroup_generators() Family ('a', 'b', 'c', 'd')
>>> from sage.all import * >>> S.semigroup_generators() Family ('a', 'b', 'c', 'd')
such that \(x^2 = x\) and \(x y x = xy\) for any \(x\) and \(y\) in \(S\):
sage: S('dab') 'dab' sage: S('dab') * S('acb') 'dabc'
>>> from sage.all import * >>> S('dab') 'dab' >>> S('dab') * S('acb') 'dabc'
It follows that the elements of \(S\) are strings without repetitions over the alphabet \(a\), \(b\), \(c\), \(d\):
sage: sorted(S.list()) ['a', 'ab', 'abc', 'abcd', 'abd', 'abdc', 'ac', 'acb', 'acbd', 'acd', 'acdb', 'ad', 'adb', 'adbc', 'adc', 'adcb', 'b', 'ba', 'bac', 'bacd', 'bad', 'badc', 'bc', 'bca', 'bcad', 'bcd', 'bcda', 'bd', 'bda', 'bdac', 'bdc', 'bdca', 'c', 'ca', 'cab', 'cabd', 'cad', 'cadb', 'cb', 'cba', 'cbad', 'cbd', 'cbda', 'cd', 'cda', 'cdab', 'cdb', 'cdba', 'd', 'da', 'dab', 'dabc', 'dac', 'dacb', 'db', 'dba', 'dbac', 'dbc', 'dbca', 'dc', 'dca', 'dcab', 'dcb', 'dcba']
>>> from sage.all import * >>> sorted(S.list()) ['a', 'ab', 'abc', 'abcd', 'abd', 'abdc', 'ac', 'acb', 'acbd', 'acd', 'acdb', 'ad', 'adb', 'adbc', 'adc', 'adcb', 'b', 'ba', 'bac', 'bacd', 'bad', 'badc', 'bc', 'bca', 'bcad', 'bcd', 'bcda', 'bd', 'bda', 'bdac', 'bdc', 'bdca', 'c', 'ca', 'cab', 'cabd', 'cad', 'cadb', 'cb', 'cba', 'cbad', 'cbd', 'cbda', 'cd', 'cda', 'cdab', 'cdb', 'cdba', 'd', 'da', 'dab', 'dabc', 'dac', 'dacb', 'db', 'dba', 'dbac', 'dbc', 'dbca', 'dc', 'dca', 'dcab', 'dcb', 'dcba']
It also follows that there are finitely many of them:
sage: S.cardinality() 64
>>> from sage.all import * >>> S.cardinality() 64
Indeed:
sage: 4 * ( 1 + 3 * (1 + 2 * (1 + 1))) 64
>>> from sage.all import * >>> Integer(4) * ( Integer(1) + Integer(3) * (Integer(1) + Integer(2) * (Integer(1) + Integer(1)))) 64
As expected, all the elements of \(S\) are idempotents:
sage: all( x.is_idempotent() for x in S ) True
>>> from sage.all import * >>> all( x.is_idempotent() for x in S ) True
Now, let us look at the structure of the semigroup:
sage: S = FiniteSemigroups().example(alphabet = ('a','b','c')) sage: S.cayley_graph(side="left", simple=True).plot() # needs sage.graphs sage.plot Graphics object consisting of 60 graphics primitives sage: S.j_transversal_of_idempotents() # random (arbitrary choice) # needs sage.graphs ['acb', 'ac', 'ab', 'bc', 'a', 'c', 'b']
>>> from sage.all import * >>> S = FiniteSemigroups().example(alphabet = ('a','b','c')) >>> S.cayley_graph(side="left", simple=True).plot() # needs sage.graphs sage.plot Graphics object consisting of 60 graphics primitives >>> S.j_transversal_of_idempotents() # random (arbitrary choice) # needs sage.graphs ['acb', 'ac', 'ab', 'bc', 'a', 'c', 'b']
We conclude by running systematic tests on this semigroup:
sage: TestSuite(S).run(verbose = True) running ._test_an_element() . . . pass running ._test_associativity() . . . 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_enumerated_set_contains() . . . pass running ._test_enumerated_set_iter_cardinality() . . . pass running ._test_enumerated_set_iter_list() . . . pass running ._test_eq() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass running ._test_some_elements() . . . pass
>>> from sage.all import * >>> TestSuite(S).run(verbose = True) running ._test_an_element() . . . pass running ._test_associativity() . . . 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_enumerated_set_contains() . . . pass running ._test_enumerated_set_iter_cardinality() . . . pass running ._test_enumerated_set_iter_list() . . . 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[source]#
Bases:
ElementWrapper
- wrapped_class#
alias of
str
- an_element()[source]#
Returns an element of the semigroup.
EXAMPLES:
sage: S = FiniteSemigroups().example() sage: S.an_element() 'cdab' sage: S = FiniteSemigroups().example(("b")) sage: S.an_element() 'b'
>>> from sage.all import * >>> S = FiniteSemigroups().example() >>> S.an_element() 'cdab' >>> S = FiniteSemigroups().example(("b")) >>> S.an_element() 'b'
- product(x, y)[source]#
Returns the product of two elements of the semigroup.
EXAMPLES:
sage: S = FiniteSemigroups().example() sage: S('a') * S('b') 'ab' sage: S('a') * S('b') * S('a') 'ab' sage: S('a') * S('a') 'a'
>>> from sage.all import * >>> S = FiniteSemigroups().example() >>> S('a') * S('b') 'ab' >>> S('a') * S('b') * S('a') 'ab' >>> S('a') * S('a') 'a'
- semigroup_generators()[source]#
Returns the generators of the semigroup.
EXAMPLES:
sage: S = FiniteSemigroups().example(alphabet=('x','y')) sage: S.semigroup_generators() Family ('x', 'y')
>>> from sage.all import * >>> S = FiniteSemigroups().example(alphabet=('x','y')) >>> S.semigroup_generators() Family ('x', 'y')