def test_printing():
R = QQ.old_poly_ring(x)
assert str(homomorphism(R.free_module(1), R.free_module(1), [0])) == \
'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1'
assert str(homomorphism(R.free_module(2), R.free_module(2), [0, 0])) == \
'Matrix([ \n[0, 0], : QQ[x]**2 -> QQ[x]**2\n[0, 0]]) '
assert str(homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])) == \
'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1/<[x]>'
assert str(R.free_module(0).identity_hom()) == 'Matrix(0, 0, []) : QQ[x]**0 -> QQ[x]**0'
def test_printing():
R = QQ.old_poly_ring(x)
assert str(homomorphism(R.free_module(1), R.free_module(1), [0])) == \
'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1'
assert str(homomorphism(R.free_module(2), R.free_module(2), [0, 0])) == \
'Matrix([ \n[0, 0], : QQ[x]**2 -> QQ[x]**2\n[0, 0]]) '
assert str(homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])) == \
'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1/<[x]>'
assert str(R.free_module(0).identity_hom()) == 'Matrix(0, 0, []) : QQ[x]**0 -> QQ[x]**0'
def test_operations():
F = QQ.old_poly_ring(x).free_module(2)
G = QQ.old_poly_ring(x).free_module(3)
f = F.identity_hom()
g = homomorphism(F, F, [0, [1, x]])
h = homomorphism(F, F, [[1, 0], 0])
i = homomorphism(F, G, [[1, 0, 0], [0, 1, 0]])
assert f == f
assert f != g
assert f != i
assert (f != F.identity_hom()) is False
assert 2*f == f*2 == homomorphism(F, F, [[2, 0], [0, 2]])
assert f/2 == homomorphism(F, F, [[S.Half, 0], [0, S.Half]])
assert f + g == homomorphism(F, F, [[1, 0], [1, x + 1]])
assert f - g == homomorphism(F, F, [[1, 0], [-1, 1 - x]])
assert f*g == g == g*f
assert h*g == homomorphism(F, F, [0, [1, 0]])
assert g*h == homomorphism(F, F, [0, 0])
assert i*f == i
assert f([1, 2]) == [1, 2]
assert g([1, 2]) == [2, 2*x]
assert i.restrict_domain(F.submodule([x, x]))([x, x]) == i([x, x])
h1 = h.quotient_domain(F.submodule([0, 1]))
assert h1([1, 0]) == h([1, 0])
assert h1.restrict_domain(h1.domain.submodule([x, 0]))([x, 0]) == h([x, 0])
raises(TypeError, lambda: f/g)
raises(TypeError, lambda: f + 1)
raises(TypeError, lambda: f + i)
raises(TypeError, lambda: f - 1)
raises(TypeError, lambda: f*i)
def test_operations():
F = QQ.old_poly_ring(x).free_module(2)
G = QQ.old_poly_ring(x).free_module(3)
f = F.identity_hom()
g = homomorphism(F, F, [0, [1, x]])
h = homomorphism(F, F, [[1, 0], 0])
i = homomorphism(F, G, [[1, 0, 0], [0, 1, 0]])
assert f == f
assert f != g
assert f != i
assert (f != F.identity_hom()) is False
assert 2*f == f*2 == homomorphism(F, F, [[2, 0], [0, 2]])
assert f/2 == homomorphism(F, F, [[S(1)/2, 0], [0, S(1)/2]])
assert f + g == homomorphism(F, F, [[1, 0], [1, x + 1]])
assert f - g == homomorphism(F, F, [[1, 0], [-1, 1 - x]])
assert f*g == g == g*f
assert h*g == homomorphism(F, F, [0, [1, 0]])
assert g*h == homomorphism(F, F, [0, 0])
assert i*f == i
assert f([1, 2]) == [1, 2]
assert g([1, 2]) == [2, 2*x]
assert i.restrict_domain(F.submodule([x, x]))([x, x]) == i([x, x])
h1 = h.quotient_domain(F.submodule([0, 1]))
assert h1([1, 0]) == h([1, 0])
assert h1.restrict_domain(h1.domain.submodule([x, 0]))([x, 0]) == h([x, 0])
raises(TypeError, lambda: f/g)
raises(TypeError, lambda: f + 1)
raises(TypeError, lambda: f + i)
raises(TypeError, lambda: f - 1)
raises(TypeError, lambda: f*i)
def test_Modules():
from sympy.polys.domains import QQ
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
assert latex(F) == r"{\mathbb{Q}\left[x, y\right]}^{2}"
assert latex(M) == \
r"\left< {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right>"
I = R.ideal(x**2, y)
assert latex(I) == r"\left< {x^{2}},{y} \right>"
Q = F / M
assert latex(
Q
) == r"\frac{{\mathbb{Q}\left[x, y\right]}^{2}}{\left< {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right>}"
assert latex(Q.submodule([1, x**3/2], [2, y])) == \
r"\left< {{\left[ {1},{\frac{x^{3}}{2}} \right]} + {\left< {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right>}},{{\left[ {2},{y} \right]} + {\left< {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right>}} \right>"
h = homomorphism(
QQ.old_poly_ring(x).free_module(2),
QQ.old_poly_ring(x).free_module(2), [0, 0])
assert latex(
h
) == r"{\left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]} : {{\mathbb{Q}\left[x\right]}^{2}} \to {{\mathbb{Q}\left[x\right]}^{2}}"
def test_Modules():
from sympy.polys.domains import QQ
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x ** 2])
assert latex(F) == r"{\mathbb{Q}\left[x, y\right]}^{2}"
assert latex(M) == r"\left< {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right>"
I = R.ideal(x ** 2, y)
assert latex(I) == r"\left< {x^{2}},{y} \right>"
Q = F / M
assert (
latex(Q)
== r"\frac{{\mathbb{Q}\left[x, y\right]}^{2}}{\left< {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right>}"
)
assert (
latex(Q.submodule([1, x ** 3 / 2], [2, y]))
== r"\left< {{\left[ {1},{\frac{x^{3}}{2}} \right]} + {\left< {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right>}},{{\left[ {2},{y} \right]} + {\left< {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right>}} \right>"
)
h = homomorphism(QQ.old_poly_ring(x).free_module(2), QQ.old_poly_ring(x).free_module(2), [0, 0])
assert (
latex(h)
== r"{\left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]} : {{\mathbb{Q}\left[x\right]}^{2}} \to {{\mathbb{Q}\left[x\right]}^{2}}"
)
def test_properties():
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
h = homomorphism(F, F, [[x, 0], [y, 0]])
assert h.kernel() == F.submodule([-y, x])
assert h.image() == F.submodule([x, 0], [y, 0])
assert not h.is_injective()
assert not h.is_surjective()
assert h.restrict_codomain(h.image()).is_surjective()
assert h.restrict_domain(F.submodule([1, 0])).is_injective()
assert h.quotient_domain(
h.kernel()).restrict_codomain(h.image()).is_isomorphism()
R2 = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1]
F = R2.free_module(2)
h = homomorphism(F, F, [[x, 0], [y, y + 1]])
assert h.is_isomorphism()
def test_properties():
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
h = homomorphism(F, F, [[x, 0], [y, 0]])
assert h.kernel() == F.submodule([-y, x])
assert h.image() == F.submodule([x, 0], [y, 0])
assert not h.is_injective()
assert not h.is_surjective()
assert h.restrict_codomain(h.image()).is_surjective()
assert h.restrict_domain(F.submodule([1, 0])).is_injective()
assert h.quotient_domain(
h.kernel()).restrict_codomain(h.image()).is_isomorphism()
R2 = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1]
F = R2.free_module(2)
h = homomorphism(F, F, [[x, 0], [y, y + 1]])
assert h.is_isomorphism()
def test_creation():
F = QQ.old_poly_ring(x).free_module(3)
G = QQ.old_poly_ring(x).free_module(2)
SM = F.submodule([1, 1, 1])
Q = F / SM
SQ = Q.submodule([1, 0, 0])
matrix = [[1, 0], [0, 1], [-1, -1]]
h = homomorphism(F, G, matrix)
h2 = homomorphism(Q, G, matrix)
assert h.quotient_domain(SM) == h2
raises(ValueError, lambda: h.quotient_domain(F.submodule([1, 0, 0])))
assert h2.restrict_domain(SQ) == homomorphism(SQ, G, matrix)
raises(ValueError, lambda: h.restrict_domain(G))
raises(ValueError, lambda: h.restrict_codomain(G.submodule([1, 0])))
raises(ValueError, lambda: h.quotient_codomain(F))
im = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
for M in [F, SM, Q, SQ]:
assert M.identity_hom() == homomorphism(M, M, im)
assert SM.inclusion_hom() == homomorphism(SM, F, im)
assert SQ.inclusion_hom() == homomorphism(SQ, Q, im)
assert Q.quotient_hom() == homomorphism(F, Q, im)
assert SQ.quotient_hom() == homomorphism(SQ.base, SQ, im)
class conv(object):
def convert(x, y=None):
return x
class dummy(object):
container = conv()
def submodule(*args):
return None
raises(TypeError, lambda: homomorphism(dummy(), G, matrix))
raises(TypeError, lambda: homomorphism(F, dummy(), matrix))
raises(
ValueError, lambda: homomorphism(QQ.old_poly_ring(x, y).free_module(3), G, matrix))
raises(ValueError, lambda: homomorphism(F, G, [0, 0]))
def test_creation():
F = QQ.old_poly_ring(x).free_module(3)
G = QQ.old_poly_ring(x).free_module(2)
SM = F.submodule([1, 1, 1])
Q = F / SM
SQ = Q.submodule([1, 0, 0])
matrix = [[1, 0], [0, 1], [-1, -1]]
h = homomorphism(F, G, matrix)
h2 = homomorphism(Q, G, matrix)
assert h.quotient_domain(SM) == h2
raises(ValueError, lambda: h.quotient_domain(F.submodule([1, 0, 0])))
assert h2.restrict_domain(SQ) == homomorphism(SQ, G, matrix)
raises(ValueError, lambda: h.restrict_domain(G))
raises(ValueError, lambda: h.restrict_codomain(G.submodule([1, 0])))
raises(ValueError, lambda: h.quotient_codomain(F))
im = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
for M in [F, SM, Q, SQ]:
assert M.identity_hom() == homomorphism(M, M, im)
assert SM.inclusion_hom() == homomorphism(SM, F, im)
assert SQ.inclusion_hom() == homomorphism(SQ, Q, im)
assert Q.quotient_hom() == homomorphism(F, Q, im)
assert SQ.quotient_hom() == homomorphism(SQ.base, SQ, im)
class conv(object):
def convert(x, y=None):
return x
class dummy(object):
container = conv()
def submodule(*args):
return None
raises(TypeError, lambda: homomorphism(dummy(), G, matrix))
raises(TypeError, lambda: homomorphism(F, dummy(), matrix))
raises(
ValueError, lambda: homomorphism(QQ.old_poly_ring(x, y).free_module(3), G, matrix))
raises(ValueError, lambda: homomorphism(F, G, [0, 0]))
