def _richcmp_(self, other, op):
"""
Compare two linear expressions.
INPUT:
- ``other`` -- another linear expression (will be enforced by
the coercion framework)
EXAMPLES::
sage: from sage.geometry.linear_expression import LinearExpressionModule
sage: L.<x> = LinearExpressionModule(QQ)
sage: x == L([0, 1])
True
sage: x == x + 1
False
sage: M.<x> = LinearExpressionModule(ZZ)
sage: L.gen(0) == M.gen(0) # because there is a conversion
True
sage: L.gen(0) == L(M.gen(0)) # this is the conversion
True
sage: x == 'test'
False
"""
return richcmp((self._coeffs, self._const),
(other._coeffs, other._const), op)
def _richcmp_(self, other, op):
"""
EXAMPLES::
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1])
sage: D = crystals.FastRankTwo(['B',2],shape=[2,1])
sage: C(0) == C(0)
True
sage: C(1) == C(0)
False
sage: C(0) == D(0)
False
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1])
sage: D = crystals.FastRankTwo(['B',2],shape=[2,1])
sage: C(0) != C(0)
False
sage: C(1) != C(0)
True
sage: C(0) != D(0)
True
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1])
sage: C(1) < C(2)
True
sage: C(2) < C(1)
False
sage: C(2) > C(1)
True
sage: C(1) <= C(1)
True
"""
return richcmp(self.value, other.value, op)
def _richcmp_(self, other, op):
"""
EXAMPLES:
For == operator::
sage: A = crystals.KirillovReshetikhin(['C',2,1], 1,2).affinization()
sage: S = A.subcrystal(max_depth=2)
sage: sorted(S)
[[[1, 1]](-1),
[[1, 2]](-1),
[](0),
[[1, 1]](0),
[[1, 2]](0),
[[1, -2]](0),
[[2, 2]](0),
[[2, -1]](1),
[[-1, -1]](1),
[](1),
[[-2, -1]](1),
[[-1, -1]](2)]
For != operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]!=S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value!=S[j].value])
True
For < operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]<S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value<S[j].value])
True
For <= operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]<=S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value<=S[j].value])
True
For > operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]>S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value>S[j].value])
True
For >= operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]>=S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value>=S[j].value])
True
"""
return richcmp(self.value, other.value, op)
def _richcmp_(self, other, op):
"""
EXAMPLES:
For == operator::
sage: A = crystals.KirillovReshetikhin(['C',2,1], 1,2).affinization()
sage: S = A.subcrystal(max_depth=2)
sage: sorted(S)
[[[1, 1]](-1),
[[1, 2]](-1),
[](0),
[[1, 1]](0),
[[1, 2]](0),
[[1, -2]](0),
[[2, 2]](0),
[[2, -1]](1),
[[-1, -1]](1),
[](1),
[[-2, -1]](1),
[[-1, -1]](2)]
For != operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]!=S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value!=S[j].value])
True
For < operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]<S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value<S[j].value])
True
For <= operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]<=S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value<=S[j].value])
True
For > operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]>S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value>S[j].value])
True
For >= operator::
sage: ([(i,j) for i in range(len(S)) for j in range(len(S)) if S[i]>=S[j]]
....: == [(i,j) for i in range(len(S)) for j in range(len(S)) if
....: S[i].value>=S[j].value])
True
"""
return richcmp(self.value, other.value, op)
def _richcmp_(self, right, op):
"""
Compare the cusps ``self`` and ``right``.
Comparison is as for rational numbers, except with the cusp oo
greater than everything but itself.
The ordering in comparison is only really meaningful for infinity
or elements that coerce to the rationals.
EXAMPLES::
sage: Cusp(2/3) == Cusp(oo)
False
sage: Cusp(2/3) < Cusp(oo)
True
sage: Cusp(2/3)> Cusp(oo)
False
sage: Cusp(2/3) > Cusp(5/2)
False
sage: Cusp(2/3) < Cusp(5/2)
True
sage: Cusp(2/3) == Cusp(5/2)
False
sage: Cusp(oo) == Cusp(oo)
True
sage: 19/3 < Cusp(oo)
True
sage: Cusp(oo) < 19/3
False
sage: Cusp(2/3) < Cusp(11/7)
True
sage: Cusp(11/7) < Cusp(2/3)
False
sage: 2 < Cusp(3)
True
"""
if not self.__b:
s = Infinity
else:
s = self._rational_()
if not right.__b:
o = Infinity
else:
o = right._rational_()
return richcmp(s, o, op)
def _richcmp_(self, other, op):
"""
Elements of this crystal are compared using the comparison in
the underlying classical crystal.
Non elements of the crystal are not comparable with elements of the
crystal, so we return ``NotImplemented``.
EXAMPLES::
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: b = K(rows=[[1]])
sage: c = K(rows=[[2]])
sage: b == c
False
sage: b == b
True
sage: b != c
True
sage: b != b
False
sage: c < b
False
sage: b < b
False
sage: b < c
True
sage: b > c
False
sage: b > b
False
sage: c > b
True
sage: b <= c
True
sage: b <= b
True
sage: c <= b
False
sage: c >= b
True
sage: b >= b
True
sage: b >= c
False
"""
return richcmp(self.value, other.value, op)
def _richcmp_(left, right, op):
"""
Compare two free algebra elements with the same parents.
The ordering is the one on the underlying sorted list of
(monomial,coefficients) pairs.
EXAMPLES::
sage: R.<x,y> = FreeAlgebra(QQ,2)
sage: x < y
True
sage: x * y < y * x
True
sage: y * x < x * y
False
"""
v = sorted(left._monomial_coefficients.items())
w = sorted(right._monomial_coefficients.items())
return richcmp(v, w, op)
def _richcmp_(self, right, op):
"""
Compare the cusps ``self`` and ``right``.
Comparison is as for elements in the number field, except with
the cusp oo which is greater than everything but itself.
The ordering in comparison is only really meaningful for infinity.
EXAMPLES::
sage: k.<a> = NumberField(x^3 + x + 1)
sage: kCusps = NFCusps(k)
Comparing with infinity::
sage: c = kCusps((a,2))
sage: d = kCusps(oo)
sage: c < d
True
sage: kCusps(oo) < d
False
Comparison as elements of the number field::
sage: kCusps(2/3) < kCusps(5/2)
False
sage: k(2/3) < k(5/2)
False
"""
if self.__b.is_zero():
if right.__b.is_zero():
return rich_to_bool(op, 0)
else:
return rich_to_bool(op, 1)
else:
if right.__b.is_zero():
return rich_to_bool(op, -1)
else:
return richcmp(self._number_field_element_(),
right._number_field_element_(), op)
def _richcmp_(self, right, op):
"""
Compare self and right.
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; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: x = Q.0; x
(1, 0)
sage: y = Q.1; y
(0, 1)
sage: x == y
False
sage: x == x
True
sage: x + x == 2*x
True
"""
return richcmp(self.vector(), right.vector(), op)
def _richcmp_(self, other, op):
"""
Comparison.
TESTS::
sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization()
sage: mg = A.module_generators[0]
sage: mg == mg
True
sage: mg == mg.f(2).e(2)
True
sage: KT = crystals.TensorProductOfKirillovReshetikhinTableaux(['C',2,1], [[1,2],[2,1]])
sage: A = crystals.AffinizationOf(KT)
sage: A(KT.module_generators[3], 1).f(0) == A.module_generators[0]
True
sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization()
sage: mg = A.module_generators[0]
sage: mg != mg.f(2)
True
sage: mg != mg.f(2).e(2)
False
sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization()
sage: S = A.subcrystal(max_depth=2)
sage: sorted(S)
[[[1, 1], [2, 2]](0),
[[1, 1], [2, 3]](0),
[[1, 2], [2, 3]](0),
[[1, 1], [3, 3]](0),
[[1, 1], [2, 3]](1),
[[1, 2], [2, 3]](1),
[[1, 2], [3, 3]](1),
[[2, 2], [3, 3]](2)]
"""
return richcmp((self._m, self._b), (other._m, other._b), op)
def _richcmp_(self, other, op):
"""
Comparison.
TESTS::
sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization()
sage: mg = A.module_generators[0]
sage: mg == mg
True
sage: mg == mg.f(2).e(2)
True
sage: KT = crystals.TensorProductOfKirillovReshetikhinTableaux(['C',2,1], [[1,2],[2,1]])
sage: A = crystals.AffinizationOf(KT)
sage: A(KT.module_generators[3], 1).f(0) == A.module_generators[0]
True
sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization()
sage: mg = A.module_generators[0]
sage: mg != mg.f(2)
True
sage: mg != mg.f(2).e(2)
False
sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization()
sage: S = A.subcrystal(max_depth=2)
sage: sorted(S)
[[[1, 1], [2, 2]](0),
[[1, 1], [2, 3]](0),
[[1, 2], [2, 3]](0),
[[1, 1], [3, 3]](0),
[[1, 1], [2, 3]](1),
[[1, 2], [2, 3]](1),
[[1, 2], [3, 3]](1),
[[2, 2], [3, 3]](2)]
"""
return richcmp((self._m, self._b), (other._m, other._b), op)
def _richcmp_(self, other, op):
r"""
Comparisons
TESTS::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a == a
True
sage: a*e == a*e
True
sage: a*b*c^3*b*d == (a*b*c)*(c^2*b*d)
True
sage: a != b
True
sage: a*b != b*a
True
sage: a*b*c^3*b*d != (a*b*c)*(c^2*b*d)
False
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a < b
True
sage: a*b < b*a
True
sage: a*b < a*a
False
sage: a^2*b < a*b*b
True
sage: b > a
True
sage: a*b > b*a
False
sage: a*b > a*a
True
sage: a*b <= b*a
True
sage: a*b <= b*a
True
sage: FA = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [FA.gen(i) for i in range(5)]
sage: a == a
True
sage: a*e == e*a
True
sage: a*b*c^3*b*d == a*d*(b^2*c^2)*c
True
sage: a != b
True
sage: a*b != a*a
True
sage: a*b*c^3*b*d != a*d*(b^2*c^2)*c
False
"""
if self._monomial == other._monomial:
# Equal
return rich_to_bool(op, 0)
if op == op_EQ or op == op_NE:
# Not equal
return rich_to_bool(op, 1)
return richcmp(self.to_word_list(), other.to_word_list(), op)
def _richcmp_(self, other, op):
r"""
Comparisons
TESTS::
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a == a
True
sage: a*e == a*e
True
sage: a*b*c^3*b*d == (a*b*c)*(c^2*b*d)
True
sage: a != b
True
sage: a*b != b*a
True
sage: a*b*c^3*b*d != (a*b*c)*(c^2*b*d)
False
sage: F = FreeMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a < b
True
sage: a*b < b*a
True
sage: a*b < a*a
False
sage: a^2*b < a*b*b
True
sage: b > a
True
sage: a*b > b*a
False
sage: a*b > a*a
True
sage: a*b <= b*a
True
sage: a*b <= b*a
True
sage: FA = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [FA.gen(i) for i in range(5)]
sage: a == a
True
sage: a*e == e*a
True
sage: a*b*c^3*b*d == a*d*(b^2*c^2)*c
True
sage: a != b
True
sage: a*b != a*a
True
sage: a*b*c^3*b*d != a*d*(b^2*c^2)*c
False
"""
if self._monomial == other._monomial:
# Equal
return rich_to_bool(op, 0)
if op == op_EQ or op == op_NE:
# Not equal
return rich_to_bool(op, 1)
return richcmp(self.to_word_list(), other.to_word_list(), op)
