def transitive(self, other=None):
r"""
TBD
"""
if other is None:
raise ValueError, "Enter a reduction map as a parameter"
elif self.get_B() != other.get_A():
raise ValueError, "These are not good candidate sets for reduction mapping"
else:
dic_h = dict()
dic_h0 = dict()
A = self.get_A()
B = self.get_B()
C = other.get_B()
f = self.get_SRWP()
f0 = self.get_SPWP()
g = other.get_SRWP()
g0 = other.get_SPWP()
for elm in A:
if elm in fixed_points(f):
if f0(elm) in fixed_points(g):
dic_h0[elm] = g0(f0(elm))
dic_h[elm] = elm
else:
dic_h[elm] = inverse(f0)(g(f0(elm)))
else:
dic_h[elm] = f(elm)
h = FiniteSetMaps(A, A).from_dict(dic_h)
h0 = FiniteSetMaps(CombinatorialScalarWrapper(dic_h0.keys()),
C).from_dict(dic_h0)
return ReductionMaps(A, C, h, h0)
def transitive(self,other=None):
r"""
TBD
"""
if other is None:
raise ValueError, "Enter a reduction map as a parameter"
elif self.get_B() != other.get_A():
raise ValueError, "These are not good candidate sets for reduction mapping"
else:
dic_h = dict()
dic_h0 = dict()
A = self.get_A()
B = self.get_B()
C = other.get_B()
f = self.get_SRWP()
f0 = self.get_SPWP()
g = other.get_SRWP()
g0 = other.get_SPWP()
for elm in A:
if elm in fixed_points(f):
if f0(elm) in fixed_points(g):
dic_h0[elm] = g0(f0(elm))
dic_h[elm] = elm
else:
dic_h[elm] = inverse(f0)(g(f0(elm)))
else:
dic_h[elm] = f(elm)
h = FiniteSetMaps(A,A).from_dict(dic_h)
h0 = FiniteSetMaps(CombinatorialScalarWrapper(dic_h0.keys()),C).from_dict(dic_h0)
return ReductionMaps(A,C,h,h0)
def __init__(self, A, B, f, f0):
r"""
TBD
"""
if type(A) != CombinatorialScalarWrapper:
raise ValueError, "The first input must be a Combinatorial Scalar Wrapper"
elif type(B) != CombinatorialScalarWrapper:
raise ValueError, "The second input must be a Combinatorial Scalar Wrapper"
elif not (bool(
A.get_generating_function() == B.get_generating_function())):
raise ValueError, "The generating functions of the scalars are not equal"
elif type(f) != FiniteSetMap_Set and type(f) != FiniteSetEndoMap_Set:
raise ValueError, "The third input must be a map in FiniteSetMaps"
elif type(f0) != FiniteSetMap_Set and type(f0) != FiniteSetEndoMap_Set:
raise ValueError, "The fourth input must be a map in FiniteSetMaps"
elif set(f.domain()) != set(A):
raise ValueError, "The third input must have domain of first input"
elif set(f0.domain()) != set(fixed_points(f)):
raise ValueError, "The fourth input must have domain of fixed points of third the input"
elif set(f0.image_set()) != set(B):
raise ValueError, "The fourth input must have image of second input"
elif not (is_SRWP_involution(f)):
raise ValueError, "The third input must be an SRWP involution"
elif not (is_SPWP_bijection(f0)):
raise ValueError, "The fourth input must be an SPWP bijection"
else:
self._A = A
self._B = B
self._f = f
self._f0 = f0
def __init__(self, A,B,f,f0):
r"""
TBD
"""
if type(A)!=CombinatorialScalarWrapper:
raise ValueError, "The first input must be a Combinatorial Scalar Wrapper"
elif type(B)!=CombinatorialScalarWrapper:
raise ValueError, "The second input must be a Combinatorial Scalar Wrapper"
elif not(bool(A.get_generating_function()==B.get_generating_function())):
raise ValueError, "The generating functions of the scalars are not equal"
elif type(f)!= FiniteSetMap_Set and type(f) != FiniteSetEndoMap_Set:
raise ValueError, "The third input must be a map in FiniteSetMaps"
elif type(f0) != FiniteSetMap_Set and type(f0) != FiniteSetEndoMap_Set:
raise ValueError, "The fourth input must be a map in FiniteSetMaps"
elif set(f.domain()) != set(A):
raise ValueError, "The third input must have domain of first input"
elif set(f0.domain()) != set(fixed_points(f)):
raise ValueError, "The fourth input must have domain of fixed points of third the input"
elif set(f0.image_set())!=set(B):
raise ValueError, "The fourth input must have image of second input"
elif not(is_SRWP_involution(f)):
raise ValueError, "The third input must be an SRWP involution"
elif not(is_SPWP_bijection(f0)):
raise ValueError, "The fourth input must be an SPWP bijection"
else:
self._A = A
self._B = B
self._f = f
self._f0 = f0
def confluence(self, other=None):
r"""
Returns the reduction of C to B, where the usage is:
r1.confluence(r2), and r1 maps A to B, B fully cancelled,
and r2 maps A to C.
"""
if other is None:
raise ValueError, "Enter a redution map as a parameter"
elif self.get_A() != other.get_A():
raise ValueError, """ "A" """ " sets have to match"
elif not (self.get_B().is_fully_cancelled()):
raise ValueError, """ "B" """ " on the left is not fully cancelled"
else:
dic_h = dict()
dic_h0 = dict()
A = self.get_A()
B = self.get_B()
C = other.get_B()
f = self.get_SRWP()
f0 = self.get_SPWP()
g = other.get_SRWP()
g0 = other.get_SPWP()
fxd_f = fixed_points(self.get_SRWP())
fxd_g = fixed_points(other.get_SRWP())
for c in C:
x = inverse(g0)(c)
while True:
if x in fxd_f: #a fixed point of h
dic_h[c] = c
dic_h0[c] = f0(x)
break
else:
x = f(x)
if x in fxd_g: #not a fixed point of h
dic_h[c] = g0(x)
break
else:
x = g(x)
h = FiniteSetMaps(C, C).from_dict(dic_h)
h0 = FiniteSetMaps(CombinatorialScalarWrapper(dic_h0.keys()),
B).from_dict(dic_h0)
return ReductionMaps(C, B, h, h0)
def confluence(self,other=None):
r"""
Returns the reduction of C to B, where the usage is:
r1.confluence(r2), and r1 maps A to B, B fully cancelled,
and r2 maps A to C.
"""
if other is None:
raise ValueError, "Enter a redution map as a parameter"
elif self.get_A() != other.get_A():
raise ValueError, """ "A" """ " sets have to match"
elif not(self.get_B().is_fully_cancelled()):
raise ValueError, """ "B" """ " on the left is not fully cancelled"
else:
dic_h = dict()
dic_h0 = dict()
A = self.get_A()
B = self.get_B()
C = other.get_B()
f = self.get_SRWP()
f0 = self.get_SPWP()
g = other.get_SRWP()
g0 = other.get_SPWP()
fxd_f = fixed_points(self.get_SRWP())
fxd_g = fixed_points(other.get_SRWP())
for c in C:
x = inverse(g0)(c)
while True:
if x in fxd_f: #a fixed point of h
dic_h[c] = c
dic_h0[c] = f0(x)
break
else:
x = f(x)
if x in fxd_g: #not a fixed point of h
dic_h[c]=g0(x)
break
else:
x = g(x)
h = FiniteSetMaps(C,C).from_dict(dic_h)
h0 = FiniteSetMaps(CombinatorialScalarWrapper(dic_h0.keys()),B).from_dict(dic_h0)
return ReductionMaps(C,B,h,h0)
def reduction_identity_matrix(mat,st=None,involution_dict=None):
r"""
When a matrix reduces to the identity, this returns
a ReductionMapDict of from a matrix to I.
"""
dim = mat.nrows()
if involution_dict is None:
fs = _involution_dict(mat)
else:
fs = involution_dict
f0s = dict()
I = identity_matrix(dim)
for i in range(dim):
for j in range(dim):
if i==j:
tmp = CombinatorialScalarWrapper(set(fixed_points(fs[i,j])))
f0s[i,j] = FiniteSetMaps(tmp,I[i,j]).from_dict({tmp.get_set().pop():CombinatorialObject(1,1)})
else:
f0s[i,j] = FiniteSetMaps(set(),set()).from_dict({})
d = dict()
for i in range(dim):
for j in range(dim):
d[i,j] = ReductionMaps(mat[i,j],I[i,j],fs[i,j],f0s[i,j])
return ReductionMapsDict(d,st)
def reduction_lemma_28_68(mat, red_adjAA_to_I, st = "an application of lemma 28, reduction_68"):
r"""
Because only one matrix here has a nontrivial SRWP map,
we need not apply the formal indexing given in the proof
of lemma 28. Simply enter a matrix (adj_AA)BA and the
reduction of adj_AA to I.
"""
d = dict()
dim = mat.nrows()
for i in range(dim):
for j in range(dim):
dic_f = dict()
dic_f0 = dict()
newset = set()
for elm in mat[i,j]:
#break tuple apart
tmp0 = elm.get_object()[0]
tmp1 = elm.get_object()[1]
tmp2 = elm.get_object()[2]
row = tmp0.get_row()
col = tmp0.get_col()
f_row_col = red_adjAA_to_I[row,col].get_SRWP()
f0_row_col = red_adjAA_to_I[row,col].get_SPWP()
#assign map
tmp = f_row_col(tmp0)
sign = tmp.get_sign()*tmp1.get_sign()*tmp2.get_sign()
weight = tmp.get_weight()*tmp1.get_weight()*tmp2.get_weight()
dic_f[elm] = CombinatorialObject((tmp,tmp1,tmp2),sign,weight)
if tmp in fixed_points(f_row_col):
tmpfxd = CombinatorialObject((f0_row_col(tmp0),tmp1,tmp2),elm.get_sign(),elm.get_weight())
dic_f0[elm] = tmpfxd
newset.add(tmpfxd)
f = FiniteSetMaps(mat[i,j],mat[i,j]).from_dict(dic_f)
f0 = FiniteSetMaps(dic_f0.keys(),newset).from_dict(dic_f0)
d[i,j] = ReductionMaps(mat[i,j],CombinatorialScalarWrapper(newset),f,f0)
return ReductionMapsDict(d,st)
