def inverse_image(self, A):
"""
Given a submodule A of the codomain of this morphism, return
the inverse image of A under this morphism.
EXAMPLES::
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]); Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi = Q.hom([0, Q.1])
sage: phi.inverse_image(Q.submodule([]))
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi.kernel()
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi.inverse_image(phi.codomain())
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi.inverse_image(Q.submodule([Q.0]))
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi.inverse_image(Q.submodule([Q.1]))
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi.inverse_image(ZZ^3)
Traceback (most recent call last):
...
TypeError: A must be a finitely generated quotient module
sage: phi.inverse_image(ZZ^3 / W.scale(2))
Traceback (most recent call last):
...
ValueError: A must be a submodule of the codomain
"""
from fgp_module import is_FGP_Module
if not is_FGP_Module(A):
raise TypeError("A must be a finitely generated quotient module")
if not A.is_submodule(self.codomain()):
raise ValueError("A must be a submodule of the codomain")
V = self._phi.inverse_image(A.V())
D = self.domain()
V = D.W() + V
return D._module_constructor(V, D.W(), check=fgp_module.DEBUG)
def inverse_image(self, A):
"""
Given a submodule A of the codomain of this morphism, return
the inverse image of A under this morphism.
EXAMPLES::
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2]); Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi = Q.hom([0, Q.1])
sage: phi.inverse_image(Q.submodule([]))
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi.kernel()
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi.inverse_image(phi.codomain())
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi.inverse_image(Q.submodule([Q.0]))
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi.inverse_image(Q.submodule([Q.1]))
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi.inverse_image(ZZ^3)
Traceback (most recent call last):
...
TypeError: A must be a finitely generated quotient module
sage: phi.inverse_image(ZZ^3 / W.scale(2))
Traceback (most recent call last):
...
ValueError: A must be a submodule of the codomain
"""
from fgp_module import is_FGP_Module
if not is_FGP_Module(A):
raise TypeError("A must be a finitely generated quotient module")
if not A.is_submodule(self.codomain()):
raise ValueError("A must be a submodule of the codomain")
V = self._phi.inverse_image(A.V())
D = self.domain()
V = D.W() + V
return D._module_constructor(V, D.W(), check=fgp_module.DEBUG)
def __call__(self, x):
"""
EXAMPLES::
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W
sage: phi = Q.hom([Q.0+3*Q.1, -Q.1]);
sage: phi(Q.0) == Q.0 + 3*Q.1
True
We compute the image of some submodules of the domain::
sage: phi(Q)
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi(Q.submodule([Q.0]))
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi(Q.submodule([Q.1]))
Finitely generated module V/W over Integer Ring with invariants (12)
sage: phi(W/W)
Finitely generated module V/W over Integer Ring with invariants ()
We try to evaluate on a module that is not a submodule of the domain, which raises a ValueError::
sage: phi(V/W.scale(2))
Traceback (most recent call last):
...
ValueError: x must be a submodule or element of the domain
We evaluate on an element of the domain that is not in the V
for the optimized representation of the domain::
sage: V = span([[1/2,0,0],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: O, X = Q.optimized()
sage: O.V()
Free module of degree 3 and rank 2 over Integer Ring
User basis matrix:
[0 0 1]
[0 2 0]
sage: phi = Q.hom([Q.0, 4*Q.1])
sage: x = Q(V.0); x
(0, 4)
sage: x == 4*Q.1
True
sage: x in O.V()
False
sage: phi(x)
(0, 4)
sage: phi(4*Q.1)
(0, 4)
sage: phi(4*Q.1) == phi(x)
True
"""
from fgp_module import is_FGP_Module
if is_FGP_Module(x):
if not x.is_submodule(self.domain()):
raise ValueError(
"x must be a submodule or element of the domain")
# perhaps can be optimized with a matrix multiply; but note
# the subtlety of optimized representations.
return self.codomain().submodule(
[self(y) for y in x.smith_form_gens()])
else:
C = self.codomain()
D = self.domain()
O, X = D.optimized()
x = D(x)
if O is D:
x = x.lift()
else:
# Now we have to transform x so that it is in the optimized representation.
x = D.V().coordinate_vector(x.lift()) * X
return C(self._phi(x))
def __call__(self, x):
"""
EXAMPLES::
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W
sage: phi = Q.hom([Q.0+3*Q.1, -Q.1]);
sage: phi(Q.0) == Q.0 + 3*Q.1
True
We compute the image of some submodules of the domain::
sage: phi(Q)
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: phi(Q.submodule([Q.0]))
Finitely generated module V/W over Integer Ring with invariants (4)
sage: phi(Q.submodule([Q.1]))
Finitely generated module V/W over Integer Ring with invariants (12)
sage: phi(W/W)
Finitely generated module V/W over Integer Ring with invariants ()
We try to evaluate on a module that is not a submodule of the domain, which raises a ValueError::
sage: phi(V/W.scale(2))
Traceback (most recent call last):
...
ValueError: x must be a submodule or element of the domain
We evaluate on an element of the domain that is not in the V
for the optimized representation of the domain::
sage: V = span([[1/2,0,0],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: O, X = Q.optimized()
sage: O.V()
Free module of degree 3 and rank 2 over Integer Ring
User basis matrix:
[0 0 1]
[0 2 0]
sage: phi = Q.hom([Q.0, 4*Q.1])
sage: x = Q(V.0); x
(0, 4)
sage: x == 4*Q.1
True
sage: x in O.V()
False
sage: phi(x)
(0, 4)
sage: phi(4*Q.1)
(0, 4)
sage: phi(4*Q.1) == phi(x)
True
"""
from fgp_module import is_FGP_Module
if is_FGP_Module(x):
if not x.is_submodule(self.domain()):
raise ValueError("x must be a submodule or element of the domain")
# perhaps can be optimized with a matrix multiply; but note
# the subtlety of optimized representations.
return self.codomain().submodule([self(y) for y in x.smith_form_gens()])
else:
C = self.codomain()
D = self.domain()
O, X = D.optimized()
x = D(x)
if O is D:
x = x.lift()
else:
# Now we have to transform x so that it is in the optimized representation.
x = D.V().coordinate_vector(x.lift()) * X
return C(self._phi(x))