Finite semigroups#

class sage.categories.finite_semigroups.FiniteSemigroups(base_category)[source]#

Bases: CategoryWithAxiom_singleton

The category of finite (multiplicative) semigroups.

A finite semigroup is a finite set endowed with an associative binary operation \(*\).

Warning

Finite semigroups in Sage used to be automatically endowed with an enumerated set structure; the default enumeration is then obtained by iteratively multiplying the semigroup generators. This forced any finite semigroup to either implement an enumeration, or provide semigroup generators; this was often inconvenient.

Instead, finite semigroups that provide a distinguished finite set of generators with semigroup_generators() should now explicitly declare themselves in the category of finitely generated semigroups:

sage: Semigroups().FinitelyGenerated()
Category of finitely generated semigroups

>>> from sage.all import *
>>> Semigroups().FinitelyGenerated()
Category of finitely generated semigroups

This is a backward incompatible change.

EXAMPLES:

sage: C = FiniteSemigroups(); C
Category of finite semigroups
sage: C.super_categories()
[Category of semigroups, Category of finite sets]
sage: sorted(C.axioms())
['Associative', 'Finite']
sage: C.example()
An example of a finite semigroup:
 the left regular band generated by ('a', 'b', 'c', 'd')

>>> from sage.all import *
>>> C = FiniteSemigroups(); C
Category of finite semigroups
>>> C.super_categories()
[Category of semigroups, Category of finite sets]
>>> sorted(C.axioms())
['Associative', 'Finite']
>>> C.example()
An example of a finite semigroup:
 the left regular band generated by ('a', 'b', 'c', 'd')
class ParentMethods[source]#

Bases: object

idempotents()[source]#

Returns the idempotents of the semigroup

EXAMPLES:

sage: S = FiniteSemigroups().example(alphabet=('x','y'))
sage: sorted(S.idempotents())
['x', 'xy', 'y', 'yx']

>>> from sage.all import *
>>> S = FiniteSemigroups().example(alphabet=('x','y'))
>>> sorted(S.idempotents())
['x', 'xy', 'y', 'yx']
j_classes()[source]#

Returns the \(J\)-classes of the semigroup.

Two elements \(u\) and \(v\) of a monoid are in the same \(J\)-class if \(u\) divides \(v\) and \(v\) divides \(u\).

OUTPUT:

All the \(J\)-classes of self, as a list of lists.

EXAMPLES:

sage: S = FiniteSemigroups().example(alphabet=('a','b', 'c'))
sage: sorted(map(sorted, S.j_classes()))                                # needs sage.graphs
[['a'], ['ab', 'ba'], ['abc', 'acb', 'bac', 'bca', 'cab', 'cba'],
 ['ac', 'ca'], ['b'], ['bc', 'cb'], ['c']]

>>> from sage.all import *
>>> S = FiniteSemigroups().example(alphabet=('a','b', 'c'))
>>> sorted(map(sorted, S.j_classes()))                                # needs sage.graphs
[['a'], ['ab', 'ba'], ['abc', 'acb', 'bac', 'bca', 'cab', 'cba'],
 ['ac', 'ca'], ['b'], ['bc', 'cb'], ['c']]
j_classes_of_idempotents()[source]#

Returns all the idempotents of self, grouped by J-class.

OUTPUT:

a list of lists.

EXAMPLES:

sage: S = FiniteSemigroups().example(alphabet=('a','b', 'c'))
sage: sorted(map(sorted, S.j_classes_of_idempotents()))                 # needs sage.graphs
[['a'], ['ab', 'ba'], ['abc', 'acb', 'bac', 'bca', 'cab', 'cba'],
 ['ac', 'ca'], ['b'], ['bc', 'cb'], ['c']]

>>> from sage.all import *
>>> S = FiniteSemigroups().example(alphabet=('a','b', 'c'))
>>> sorted(map(sorted, S.j_classes_of_idempotents()))                 # needs sage.graphs
[['a'], ['ab', 'ba'], ['abc', 'acb', 'bac', 'bca', 'cab', 'cba'],
 ['ac', 'ca'], ['b'], ['bc', 'cb'], ['c']]
j_transversal_of_idempotents()[source]#

Returns a list of one idempotent per regular J-class

EXAMPLES:

sage: S = FiniteSemigroups().example(alphabet=('a','b', 'c'))

>>> from sage.all import *
>>> S = FiniteSemigroups().example(alphabet=('a','b', 'c'))

The chosen elements depend on the order of each \(J\)-class, and that order is random when using Python 3.

sage: sorted(S.j_transversal_of_idempotents())  # random                # needs sage.graphs
['a', 'ab', 'abc', 'ac', 'b', 'c', 'cb']

>>> from sage.all import *
>>> sorted(S.j_transversal_of_idempotents())  # random                # needs sage.graphs
['a', 'ab', 'abc', 'ac', 'b', 'c', 'cb']