def cartan_matrix_as_graph(self, q=None):
from sage.graphs.graph import DiGraph
G = DiGraph()
G.add_vertices(self.idempotents())
for (e,f),coeff in self.cartan_matrix_as_table(q).iteritems():
G.add_edge(e,f, None if coeff == 1 else coeff)
return G
def j_poset(self):
r"""
Returns the j-order on the j-classes, as a poset on the
indices of the j-classes. For two such indices `i` and
`j`, and `x` and `y` in corresponding J-classes, one has:
`i \le j \Leftrightarrow x \in S y S \Leftrightarrow S x S \subset S y S`
In particular, the identity (when it exists) is the unique
maximal element of this poset.
EXAMPLES::
sage: S = Semigroups().Finite().example(alphabet=('a','b', 'c'))
sage: P = S.j_poset(); P
Finite poset containing 7 elements
sage: P.cover_relations() # random (arbitrary choice)
[['cab', 'ca'], ['cab', 'cb'], ['cab', 'ab'], ['ca', 'c'], ['ca', 'a'], ['cb', 'b'], ['cb', 'c'], ['ab', 'b'], ['ab', 'a']]
sage: len(P.cover_relations())
9
"""
from sage.graphs.graph import DiGraph
from sage.combinat.posets.posets import Poset
# This is more or less duplicating what strongly_connected_components_digraph does!!!
G = DiGraph()
G.add_vertices(self.j_classes().keys())
for (x,y,_) in self.cayley_graph_cached(side="twosided", simple=True).edge_iterator():
G.add_edge(self.j_class_index(y), self.j_class_index(x))
return Poset(G)
def cartan_matrix_as_graph(self, q=None):
from sage.graphs.graph import DiGraph
G = DiGraph()
G.add_vertices(self.idempotents())
for (e,f),coeff in self.cartan_matrix_as_table(q).iteritems():
G.add_edge(e,f, None if coeff == 1 else coeff)
return G
def quiver(self, edge_labels=True, index=False):
"""
Returns the quiver of ``self``
INPUT:
- ``edges_labels`` -- whether to use the quiver element
as label for the edges between the idempotents; if
False, this may lead to multiple edges when the monoid
is not generated by idempotents (default: True)
- ``index`` -- whether to index the vertices of the graph
by the indices of the simple modules rather than the
corresponding idempotents (default: False)
OUTPUT: the quiver of ``self``, as a graph with the
idempotents of this monoid as vertices
.. todo:: should index default to True?
.. todo:: use a meaningful example
EXAMPLES::
sage: import sage_semigroups.monoids.catalog as semigroups
sage: M = semigroups.NDPFMonoidPoset(Posets(3)[3])
sage: G = M.quiver()
sage: G
Digraph on 4 vertices
sage: G.edges()
[([1], [2], [3])]
sage: M.quiver(edge_labels=False).edges()
[([1], [2], None)]
sage: M.quiver(index=True).edges()
[(2, 1, [3])]
sage: M.quiver(index=True, edge_labels=False).edges()
[(2, 1, None)]
"""
from sage.graphs.graph import DiGraph
res = DiGraph()
if index:
res.add_vertices(self.simple_modules_index_set())
symbol = attrcall("symbol_index")
else:
res.add_vertices(self.idempotents())
symbol = attrcall("symbol")
for x in self.quiver_elements():
res.add_edge(symbol(x,"left"), symbol(x, "right"), x if edge_labels else None)
return res
def quiver(self, edge_labels=True, index=False):
"""
Returns the quiver of ``self``
INPUT:
- ``edges_labels`` -- whether to use the quiver element
as label for the edges between the idempotents; if
False, this may lead to multiple edges when the monoid
is not generated by idempotents (default: True)
- ``index`` -- whether to index the vertices of the graph
by the indices of the simple modules rather than the
corresponding idempotents (default: False)
OUTPUT: the quiver of ``self``, as a graph with the
idempotents of this monoid as vertices
.. todo:: should index default to True?
.. todo:: use a meaningful example
EXAMPLES::
sage: import sage_semigroups.monoids.catalog as semigroups
sage: M = semigroups.NDPFMonoidPoset(Posets(3)[3])
sage: G = M.quiver()
sage: G
Digraph on 4 vertices
sage: G.edges()
[([1], [2], [3])]
sage: M.quiver(edge_labels=False).edges()
[([1], [2], None)]
sage: M.quiver(index=True).edges()
[(2, 1, [3])]
sage: M.quiver(index=True, edge_labels=False).edges()
[(2, 1, None)]
"""
from sage.graphs.graph import DiGraph
res = DiGraph()
if index:
res.add_vertices(self.simple_modules_index_set())
symbol = attrcall("symbol_index")
else:
res.add_vertices(self.idempotents())
symbol = attrcall("symbol")
for x in self.quiver_elements():
res.add_edge(symbol(x,"left"), symbol(x, "right"), x if edge_labels else None)
return res
