def __init__(self,data=None,alphabet=None):
self._initial={}
self._terminal={}
letters=[]
if isinstance(data,dict):
new_data=dict()
for a in data:
letters.append(a)
if data[a][0] in new_data:
if data[a][1] in new_data[data[a][0]]:
new_data[data[a][0]][data[a][1]].append(a)
else:
new_data[data[a][0]][data[a][1]]=[a]
else:
new_data[data[a][0]]={data[a][1]:[a]}
data=new_data
elif isinstance(data,list):
new_data=dict()
for e in data:
letters.append(e[2])
if e[0] in new_data:
if e[1] in new_data[e[0]]:
new_data[e[0]][e[1]].append(e[2])
else:
new_data[e[0]][e[1]]=[e[2]]
else:
new_data[e[0]]={e[1]:[e[2]]}
data=new_data
if alphabet is None:
from inverse_alphabet import AlphabetWithInverses
alphabet = AlphabetWithInverses(self._initial.keys())
self._alphabet=alphabet
DiGraph.__init__(self,data=data,loops=True,multiedges=True,vertex_labels=True,pos=None,format=None,\
boundary=[],weighted=None,implementation='c_graph',sparse=True)
for e in self.edges():
self._initial[e[2]]=e[0]
self._terminal[e[2]]=e[1]
self._initial[alphabet.inverse_letter(e[2])]=e[1]
self._terminal[alphabet.inverse_letter(e[2])]=e[0]
def __init__(self, data=None, alphabet=None):
self._initial = {}
self._terminal = {}
letters = []
if isinstance(data, dict):
new_data = dict()
for a in data:
letters.append(a)
if data[a][0] in new_data:
if data[a][1] in new_data[data[a][0]]:
new_data[data[a][0]][data[a][1]].append(a)
else:
new_data[data[a][0]][data[a][1]] = [a]
else:
new_data[data[a][0]] = {data[a][1]: [a]}
data = new_data
elif isinstance(data, list):
new_data = dict()
for e in data:
letters.append(e[2])
if e[0] in new_data:
if e[1] in new_data[e[0]]:
new_data[e[0]][e[1]].append(e[2])
else:
new_data[e[0]][e[1]] = [e[2]]
else:
new_data[e[0]] = {e[1]: [e[2]]}
data = new_data
if alphabet is None:
from inverse_alphabet import AlphabetWithInverses
alphabet = AlphabetWithInverses(self._initial.keys())
self._alphabet = alphabet
DiGraph.__init__(self,data=data,loops=True,multiedges=True,vertex_labels=True,pos=None,format=None,\
boundary=[],weighted=None,implementation='c_graph',sparse=True)
for e in self.edges():
self._initial[e[2]] = e[0]
self._terminal[e[2]] = e[1]
self._initial[alphabet.inverse_letter(e[2])] = e[1]
self._terminal[alphabet.inverse_letter(e[2])] = e[0]
def __init__(self,data,group=None):
"""
Builds a FreeGroupAutomorphism from data.
INPUT:
- ``data`` - the data used to build the morphism
- ``group`` - an optional free group
"""
if group is not None and not isinstance(group, FreeGroup):
raise ValueError, "the group must be a Free Group"
WordMorphism.__init__(self,data)
if group is None:
A = AlphabetWithInverses(self.domain().alphabet())
F = FreeGroup(A)
else:
F = group
A = group.alphabet()
self._domain = F
self._codomain = F # unuseful... consistency with WordMorphism
for letter in self._morph.keys():
self._morph[letter]=F.reduce(self._morph[letter])
self._morph[A.inverse_letter(letter)] = F.inverse_word(self._morph[letter])
def __init__(self, data, group=None):
"""
Builds a FreeGroupAutomorphism from data.
INPUT:
- ``data`` - the data used to build the morphism
- ``group`` - an optional free group
"""
if group is not None and not isinstance(group, FreeGroup):
raise ValueError, "the group must be a Free Group"
WordMorphism.__init__(self, data)
if group is None:
A = AlphabetWithInverses(self.domain().alphabet())
F = FreeGroup(A)
else:
F = group
A = group.alphabet()
self._domain = F
self._codomain = F # unuseful... consistency with WordMorphism
for letter in self._morph.keys():
self._morph[letter] = F.reduce(self._morph[letter])
self._morph[A.inverse_letter(letter)] = F.inverse_word(
self._morph[letter])
def __init__(self, data=None, alphabet=None):
"""
INPUT:
- ``data`` -- (default None) a list or dictionary
from a list of edges: ``[initial_vertex,terminal_vertex,letter]``.
a dictionnary that maps letters of the alphabet to lists
``(initial_vertex,terminal_vertex)``
- ``alphabet`` -- (default None ) alphabet AlphabetWithInverses by default
"""
self._initial = {}
self._terminal = {}
letters = []
if isinstance(data, dict):
new_data = dict()
for a in data:
letters.append(a)
if data[a][0] in new_data:
if data[a][1] in new_data[data[a][0]]:
new_data[data[a][0]][data[a][1]].append(a)
else:
new_data[data[a][0]][data[a][1]] = [a]
else:
new_data[data[a][0]] = {data[a][1]: [a]}
data = new_data
elif isinstance(data, list):
new_data = dict()
for e in data:
letters.append(e[2])
if e[0] in new_data:
if e[1] in new_data[e[0]]:
new_data[e[0]][e[1]].append(e[2])
else:
new_data[e[0]][e[1]] = [e[2]]
else:
new_data[e[0]] = {e[1]: [e[2]]}
data = new_data
if alphabet is None:
from inverse_alphabet import AlphabetWithInverses
alphabet = AlphabetWithInverses(letters)
self._alphabet = alphabet
DiGraph.__init__(self, data=data, loops=True, multiedges=True,
vertex_labels=True, pos=None, format=None,
weighted=None,
implementation='c_graph', sparse=True)
# DiGraph.__init__(self,data=data, loops=True,multiedges=True,
# vertex_labels=True, pos=None, format=None,
# boundary=[], weighted=None,
# implementation='c_graph', sparse=True)
for e in self.edges():
self._initial[e[2]] = e[0]
self._terminal[e[2]] = e[1]
self._initial[alphabet.inverse_letter(e[2])] = e[1]
self._terminal[alphabet.inverse_letter(e[2])] = e[0]
def inverse(self):
"""A homotopy inverse of ``self``.
For ``t1=self.domain().spanning_tree()`` and
``t2=self.codomain().spanning_tree()``. The alphabet ``A`` is
made of edges not in ``t1`` identified (by their order in the
alphabets of the domain and codomain) to letters not in
``t2``. The automorphism ``phi`` of ``FreeGroup(A)`` is
defined using ``t1`` and ``t2``. The inverse map is given by
``phi.inverse()`` using ``t1`` and edges from ``t2`` are
mapped to a single point (the root of ``t1``).
In particular the inverse maps all vertices to the root of ``t1``.
WARNING:
``self`` is assumed to be a homotopy equivalence.
"""
from free_group import FreeGroup
from free_group_automorphism import FreeGroupAutomorphism
G1=self.domain()
A1=G1.alphabet()
t1=G1.spanning_tree()
G2=self.codomain()
A2=G2.alphabet()
t2=G2.spanning_tree()
A=AlphabetWithInverses(len(A1)-len(G1.vertices())+1)
F=FreeGroup(A)
map=dict()
translate=dict()
i=0
for a in A1.positive_letters():
l=len(t1[G1.initial_vertex(a)])-len(t1[G1.terminal_vertex(a)])
if (l!=1 or t1[G1.initial_vertex(a)][-1]!=A1.inverse_letter(a)) and\
(l!=-1 or t1[G1.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
map[A[i]]=self(t1[G1.initial_vertex(a)]*Word([a])*G1.reverse_path(t1[G1.terminal_vertex(a)]))
translate[A[i]]=a
translate[A.inverse_letter(A[i])]=A1.inverse_letter(a)
i+=1
rename=dict()
edge_map=dict()
i=0
for a in A2.positive_letters():
l=len(t2[G2.initial_vertex(a)])-len(t2[G2.terminal_vertex(a)])
if (l!=1 or t2[G2.initial_vertex(a)][-1]!=A2.inverse_letter(a)) and\
(l!=-1 or t2[G2.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
rename[a]=A[i]
rename[A2.inverse_letter(a)]=A.inverse_letter(A[i])
i+=1
else:
edge_map[a]=Word()
for a in map:
map[a]=F([rename[b] for b in map[a] if b in rename])
phi=FreeGroupAutomorphism(map,F)
psi=phi.inverse()
i=0
for a in A2.positive_letters():
if a not in edge_map:
result=Word()
for b in psi.image(A[i]):
c=translate[b]
result=result*t1[G1.initial_vertex(c)]*Word([c])*G1.reverse_path(t1[G1.terminal_vertex(c)])
edge_map[a]=G1.reduce_path(result)
i+=1
return GraphMap(G2,G1,edge_map)
def inverse(self):
"""A homotopy inverse of ``self``.
For ``t1=self.domain().spanning_tree()`` and
``t2=self.codomain().spanning_tree()``. The alphabet ``A`` is
made of edges not in ``t1`` identified (by their order in the
alphabets of the domain and codomain) to letters not in
``t2``. The automorphism ``phi`` of ``FreeGroup(A)`` is
defined using ``t1`` and ``t2``. The inverse map is given by
``phi.inverse()`` using ``t1`` and edges from ``t2`` are
mapped to a single point (the root of ``t1``).
In particular the inverse maps all vertices to the root of ``t1``.
OUTPUT:
A homotopy inverse of ``self``.
WARNING:
``self`` is assumed to be a homotopy equivalence.
EXAMPLES::
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: A = AlphabetWithInverses(2)
sage: H = GraphWithInverses.rose_graph(A)
sage: f = GraphMap(G,H,"a->ab,b->b,c->B")
sage: print f.inverse()
Graph map:
Graph with inverses: a: 0->0, b: 0->0
Graph with inverses: a: 0->0, b: 0->1, c: 1->1
edge map: a->abcB, b->bCB
"""
from free_group import FreeGroup
from free_group_automorphism import FreeGroupAutomorphism
g1 = self.domain()
a1 = g1.alphabet()
t1 = g1.spanning_tree()
g2 = self.codomain()
a2 = g2.alphabet()
t2 = g2.spanning_tree()
A = AlphabetWithInverses(len(a1) - len(g1.vertices()) + 1)
f = FreeGroup(A)
map = dict()
translate = dict()
i = 0
for a in a1.positive_letters():
l = len(t1[g1.initial_vertex(a)]) - len(t1[g1.terminal_vertex(a)])
if (l != 1 or t1[g1.initial_vertex(a)][-1] != a1.inverse_letter(
a)) and (l != -1 or t1[g1.terminal_vertex(a)][-1] != a):
# a is not in the spanning tree
map[A[i]] = self(
t1[g1.initial_vertex(a)] * Word([a]) * g1.reverse_path(
t1[g1.terminal_vertex(a)]))
translate[A[i]] = a
translate[A.inverse_letter(A[i])] = a1.inverse_letter(a)
i += 1
rename = dict()
edge_map = dict()
i = 0
for a in a2.positive_letters():
l = len(t2[g2.initial_vertex(a)]) - len(t2[g2.terminal_vertex(a)])
if (l != 1 or t2[g2.initial_vertex(a)][-1] != a2.inverse_letter(
a)) and (l != -1 or t2[g2.terminal_vertex(a)][-1] != a):
# a is not in the spanning tree
rename[a] = A[i]
rename[a2.inverse_letter(a)] = A.inverse_letter(A[i])
i += 1
else:
edge_map[a] = Word()
for a in map:
map[a] = f([rename[b] for b in map[a] if b in rename])
phi = FreeGroupAutomorphism(map, f)
psi = phi.inverse()
i = 0
for a in a2.positive_letters():
if a not in edge_map:
result = Word()
for b in psi.image(A[i]):
c = translate[b]
result = result * t1[g1.initial_vertex(c)] * Word(
[c]) * g1.reverse_path(t1[g1.terminal_vertex(c)])
edge_map[a] = g1.reduce_path(result)
i += 1
return GraphMap(g2, g1, edge_map)
def inverse(self):
"""A homotopy inverse of ``self``.
For ``t1=self.domain().spanning_tree()`` and
``t2=self.codomain().spanning_tree()``. The alphabet ``A`` is
made of edges not in ``t1`` identified (by their order in the
alphabets of the domain and codomain) to letters not in
``t2``. The automorphism ``phi`` of ``FreeGroup(A)`` is
defined using ``t1`` and ``t2``. The inverse map is given by
``phi.inverse()`` using ``t1`` and edges from ``t2`` are
mapped to a single point (the root of ``t1``).
In particular the inverse maps all vertices to the root of ``t1``.
WARNING:
``self`` is assumed to be a homotopy equivalence.
"""
from free_group import FreeGroup
from free_group_automorphism import FreeGroupAutomorphism
G1 = self.domain()
A1 = G1.alphabet()
t1 = G1.spanning_tree()
G2 = self.codomain()
A2 = G2.alphabet()
t2 = G2.spanning_tree()
A = AlphabetWithInverses(len(A1) - len(G1.vertices()) + 1)
F = FreeGroup(A)
map = dict()
translate = dict()
i = 0
for a in A1.positive_letters():
l = len(t1[G1.initial_vertex(a)]) - len(t1[G1.terminal_vertex(a)])
if (l!=1 or t1[G1.initial_vertex(a)][-1]!=A1.inverse_letter(a)) and\
(l!=-1 or t1[G1.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
map[A[i]] = self(t1[G1.initial_vertex(a)] * Word([a]) *
G1.reverse_path(t1[G1.terminal_vertex(a)]))
translate[A[i]] = a
translate[A.inverse_letter(A[i])] = A1.inverse_letter(a)
i += 1
rename = dict()
edge_map = dict()
i = 0
for a in A2.positive_letters():
l = len(t2[G2.initial_vertex(a)]) - len(t2[G2.terminal_vertex(a)])
if (l!=1 or t2[G2.initial_vertex(a)][-1]!=A2.inverse_letter(a)) and\
(l!=-1 or t2[G2.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
rename[a] = A[i]
rename[A2.inverse_letter(a)] = A.inverse_letter(A[i])
i += 1
else:
edge_map[a] = Word()
for a in map:
map[a] = F([rename[b] for b in map[a] if b in rename])
phi = FreeGroupAutomorphism(map, F)
psi = phi.inverse()
i = 0
for a in A2.positive_letters():
if a not in edge_map:
result = Word()
for b in psi.image(A[i]):
c = translate[b]
result = result * t1[G1.initial_vertex(c)] * Word(
[c]) * G1.reverse_path(t1[G1.terminal_vertex(c)])
edge_map[a] = G1.reduce_path(result)
i += 1
return GraphMap(G2, G1, edge_map)