Finite semigroups

class sage.categories.finite_semigroups.FiniteSemigroups(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton

The category of finite (multiplicative) semigroups.

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


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

This is a backward incompatible change.


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')
class ParentMethods

Bases: object


Returns the idempotents of the semigroup


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

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 \(u\).


All the $J$-classes of self, as a list of lists.


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

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


a list of lists.


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

Returns a list of one idempotent per regular J-class


sage: S = FiniteSemigroups().example(alphabet=('a','b', 'c'))
sage: sorted(S.j_transversal_of_idempotents()) # py2
['a', 'acb', 'b', 'ba', 'bc', 'c', 'ca']

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()) # py3 random
['a', 'ab', 'abc', 'ac', 'b', 'c', 'cb']