def green_classes(self, side = "twosided"):
r"""
INPUT:
- ``side`` -- "left", "right", or "twosided"
Depending on the value of ``side``, returns respectively
the `L`-classes, `R`-classes, or `J`-classes of ``self``,
sorted decreasingly along a linear extension of
respectively `L`, `R` or `J`-order.
EXAMPLES::
sage: M = Semigroups().Finite().example(('a','b')); M
An example of a finite semigroup: the left regular band generated by ('a', 'b')
sage: M.green_classes()
[{'a'}, {'b'}, {'ab', 'ba'}]
sage: M.green_classes(side="left")
[{'a'}, {'b'}, {'ab', 'ba'}]
sage: M.green_classes(side="right")
[{'b'}, {'a'}, {'ab'}, {'ba'}]
"""
G = self.cayley_graph_cached(side=side, simple=True).strongly_connected_components_digraph()
from sage.combinat.posets.posets import Poset
P = Poset(G, facade = True)
return P.linear_extension()
def classes(self): # Or discrete composition_series?
r"""
Return the classes of ``self``
The classes of ``self`` are the strongly connected components
of its Cayley graph. For a group, those would be the orbits.
EXAMPLES::
sage: M = Semigroups().SetsWithAction().example(); M
Representation of the monoid generated by <2,3> acting on Z/10 Z by multiplication
sage: M.classes()
[{1, 3, 9, 7}, {5}, {0}, {8, 2, 4, 6}]
This for example means that, in `\ZZ/10\ZZ`, any element of
``\{1,3,9,7\}`` is reachable from any other by successive
multiplications by `2` and `3`.
.. seealso: :meth:`cayley_graph`, :meth:`class_of`
"""
G = self.cayley_graph(simple=True).strongly_connected_components_digraph()
from sage.combinat.posets.posets import Poset
P = Poset(G, facade=True)
return P.linear_extension()
