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 set_terminal_vertex(self, e, v):
"""
Sets the terminal vertex of the edge ``e`` to the vertex
``v``.
Consistantly sets the initial vertex of the edge label by
the inverse of ``e`` to the vertex ``v``.
INPUT:
- ``e`` the edge
- ``v`` the vertex
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,0,'c']])
sage: G.set_terminal_vertex('a',1)
sage: print G
Graph with inverses: a: 0->1, b: 0->1, c: 1->0
"""
w = self.initial_vertex(e)
ww = self.terminal_vertex(e)
pe = self._alphabet.to_positive_letter(e)
if e == pe:
DiGraph.delete_edge(self, w, ww, pe)
DiGraph.add_edge(self, w, v, pe)
else:
DiGraph.delete_edge(self, ww, w, pe)
DiGraph.add_edge(self, v, w, pe)
self._terminal[e] = v
self._initial[self._alphabet.inverse_letter(e)] = v
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 set_terminal_vertex(self, e, v):
"""
Sets the terminal vertex of the edge ``e`` to the vertex
``v``.
Consistantly sets the initial vertex of the edge label by
the inverse of ``e`` to the vertex ``v``.
INPUT:
- ``e`` -- the edge
- ``v`` -- the vertex
EXAMPLES::
sage: from train_track.inverse_graph import GraphWithInverses
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,0,'c']])
sage: G.set_terminal_vertex('a',1)
sage: print(G)
a: 0->1, b: 0->1, c: 1->0
"""
w = self.initial_vertex(e)
ww = self.terminal_vertex(e)
pe = self._alphabet.to_positive_letter(e)
if e == pe:
DiGraph.delete_edge(self, w, ww, pe)
DiGraph.add_edge(self, w, v, pe)
else:
DiGraph.delete_edge(self, ww, w, pe)
DiGraph.add_edge(self, v, w, pe)
self._terminal[e] = v
self._initial[self._alphabet.inverse_letter(e)] = v
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 add_edge(self, u, v=None, label=None):
"""
Add a new edge.
The following forms are all accepted
- G.add_edge(1,2,'a')
- G.add_edge((1,2,'a'))
- G.add_edge(1,2,['a','A'])
- G.add_edge((1,2,['a','A']))
INPUT:
- ``u`` -- edge to add
- ``v`` -- (default: ``None``)
- ``label`` -- (default: ``None``) the label of the new edge
OUTPUT:
The label of the new edge.
.. WARNING::
Does not change the alphabet of ``self``. (the new label is
assumed to be already in the alphabet).
EXAMPLES::
sage: from train_track.inverse_graph import GraphWithInverses
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,0,'c']])
sage: a = G.alphabet().add_new_letter()
sage: G.add_edge(1,1,a)
'd'
sage: print(G)
a: 0->0, b: 0->1, c: 1->0, d: 1->1
"""
if label is None:
v = u[1]
label = u[2]
u = u[0]
if isinstance(label, list):
DiGraph.add_edge(self, u, v, label[0])
self._initial[label[0]] = u
self._initial[label[1]] = v
self._terminal[label[1]] = u
self._terminal[label[0]] = v
label = label[0]
else:
DiGraph.add_edge(self, u, v, label)
self._initial[label] = u
self._terminal[label] = v
inv_label = self.alphabet().inverse_letter(label)
self._initial[inv_label] = v
self._terminal[inv_label] = u
return label
def add_edge(self, u, v=None, label=None):
"""
Add a new edge.
The following forms are all accepted
- G.add_edge(1,2,'a')
- G.add_edge((1,2,'a'))
- G.add_edge(1,2,['a','A'])
- G.add_edge((1,2,['a','A']))
INPUT:
- ``u`` -- edge to add
- ``v`` -- (default: ``None``)
- ``label`` -- (default: ``None``) the label of the new edge
OUTPUT:
The label of the new edge.
.. WARNING::
Does not change the alphabet of ``self``. (the new label is
assumed to be already in the alphabet).
EXAMPLES::
sage: from train_track.inverse_graph import GraphWithInverses
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,0,'c']])
sage: a = G.alphabet().add_new_letter()
sage: G.add_edge(1,1,a)
'd'
sage: print(G)
a: 0->0, b: 0->1, c: 1->0, d: 1->1
"""
if label is None:
v = u[1]
label = u[2]
u = u[0]
if isinstance(label, list):
DiGraph.add_edge(self, u, v, label[0])
self._initial[label[0]] = u
self._initial[label[1]] = v
self._terminal[label[1]] = u
self._terminal[label[0]] = v
label = label[0]
else:
DiGraph.add_edge(self, u, v, label)
self._initial[label] = u
self._terminal[label] = v
inv_label = self.alphabet().inverse_letter(label)
self._initial[inv_label] = v
self._terminal[inv_label] = u
return label
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
def add_edge(self,u,v=None,label=None):
"""
Add a new edge.
INPUT: The following forms are all accepted
- G.add_edge(1,2,'a')
- G.add_edge((1,2,'a'))
- G.add_edge(1,2,['a','A'])
- G.add_edge((1,2,['a','A']))
OUTPUT:
the label of the new edge.
WARNING:
Does not change the alphabet of ``self``. (the new label is
assumed to be already in the alphabet).
"""
if label is None:
v=u[1]
label=u[2]
u=u[0]
if isinstance(label,list):
DiGraph.add_edge(self,u,v,label[0])
self._initial[label[0]]=u
self._initial[label[1]]=v
self._terminal[label[1]]=u
self._terminal[label[0]]=v
label=label[0]
else:
DiGraph.add_edge(self,u,v,label)
self._initial[label]=u
self._terminal[label]=v
inv_label=self.alphabet().inverse_letter(label)
self._initial[inv_label]=v
self._terminal[inv_label]=u
return label
def add_edge(self, u, v=None, label=None):
"""
Add a new edge.
INPUT: The following forms are all accepted
- G.add_edge(1,2,'a')
- G.add_edge((1,2,'a'))
- G.add_edge(1,2,['a','A'])
- G.add_edge((1,2,['a','A']))
OUTPUT:
the label of the new edge.
WARNING:
Does not change the alphabet of ``self``. (the new label is
assumed to be already in the alphabet).
"""
if label is None:
v = u[1]
label = u[2]
u = u[0]
if isinstance(label, list):
DiGraph.add_edge(self, u, v, label[0])
self._initial[label[0]] = u
self._initial[label[1]] = v
self._terminal[label[1]] = u
self._terminal[label[0]] = v
label = label[0]
else:
DiGraph.add_edge(self, u, v, label)
self._initial[label] = u
self._terminal[label] = v
inv_label = self.alphabet().inverse_letter(label)
self._initial[inv_label] = v
self._terminal[inv_label] = u
return label
def set_terminal_vertex(self,e,v):
"""
Sets the terminal vertex of the edge ``e`` to the vertex
``v``.
Consistantly sets the initial vertex of the edge label by
the inverse of ``e`` to the vertex ``v``.
"""
w=self.initial_vertex(e)
ww=self.terminal_vertex(e)
pe=self._alphabet.to_positive_letter(e)
if e==pe:
DiGraph.delete_edge(self,w,ww,pe)
DiGraph.add_edge(self,w,v,pe)
else:
DiGraph.delete_edge(self,ww,w,pe)
DiGraph.add_edge(self,v,w,pe)
self._terminal[e]=v
self._initial[self._alphabet.inverse_letter(e)]=v
def set_terminal_vertex(self, e, v):
"""
Sets the terminal vertex of the edge ``e`` to the vertex
``v``.
Consistantly sets the initial vertex of the edge label by
the inverse of ``e`` to the vertex ``v``.
"""
w = self.initial_vertex(e)
ww = self.terminal_vertex(e)
pe = self._alphabet.to_positive_letter(e)
if e == pe:
DiGraph.delete_edge(self, w, ww, pe)
DiGraph.add_edge(self, w, v, pe)
else:
DiGraph.delete_edge(self, ww, w, pe)
DiGraph.add_edge(self, v, w, pe)
self._terminal[e] = v
self._initial[self._alphabet.inverse_letter(e)] = v
