def __pow__(self, n, dummy=None):
"""
Exponentiate an endomorphism.
INPUT:
- ``n`` -- integer. The exponent.
OUTPUT:
A scheme morphism in the same endomorphism set as ``self``.
EXAMPLES::
sage: X = AffineSpace(QQ,2)
sage: id = X.identity_morphism()
sage: id^0
Scheme endomorphism of Affine Space of dimension 2 over Rational Field
Defn: Identity map
sage: id^2
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for *:
'SchemeMorphism_id' and 'SchemeMorphism_id'
"""
if not self.is_endomorphism():
raise TypeError, "self must be an endomorphism."
if n==0:
return self.domain().identity_morphism()
return generic_power(self, n)
def _pow_(self, n):
"""
Return ``self`` to the `n^{th}` power.
INPUT:
- ``n`` -- a positive integer
EXAMPLES::
sage: S = Semigroups().example("leftzero")
sage: x = S("x")
sage: x^1, x^2, x^3, x^4, x^5
('x', 'x', 'x', 'x', 'x')
sage: x^0
Traceback (most recent call last):
...
AssertionError
TESTS::
sage: x._pow_(17)
'x'
"""
assert n > 0
return generic_power(self, n)
def __pow__(self, n, dummy=None):
"""
Exponentiate an endomorphism.
INPUT:
- ``n`` -- integer. The exponent.
OUTPUT:
A composite map that belongs to the same endomorphism set as ``self``.
EXAMPLES::
sage: X = AffineSpace(QQ,2)
sage: id = X.identity_morphism()
sage: id^0
Scheme endomorphism of Affine Space of dimension 2 over Rational Field
Defn: Identity map
sage: id^2
Scheme endomorphism of Affine Space of dimension 2 over Rational Field
Defn: Identity map
"""
if not self.is_endomorphism():
raise TypeError, "self must be an endomorphism."
if n==0:
return self.domain().identity_morphism()
return generic_power(self, n)
def __pow__(self, n):
"""
returns self^n, where other is an integer.
TESTS::
sage: f = FreeGroupAutomorphism('a->ab,b->A')
sage: f**-3
Automorphism of the Free group over ['a', 'b']: a->bAB,b->baBAB
"""
if n > 0:
from sage.structure.element import generic_power
return generic_power(self,n)
elif n < 0:
from sage.structure.element import generic_power
return generic_power(self.inverse(), -n)
else:
return self.domain().identity_automorphism()
def __pow__(self, n):
r"""
INPUTS:
- ``n``: a non negative integer
Returns ``self`` to the `n^{th}` power.
EXAMPLES::
sage: S = Monoids().example()
sage: x = S("aa")
sage: x^0, x^1, x^2, x^3, x^4, x^5
('', 'aa', 'aaaa', 'aaaaaa', 'aaaaaaaa', 'aaaaaaaaaa')
"""
if not n: # FIXME: why do we need to do that?
return self.parent().one()
return generic_power(self, n, self.parent().one())
def __pow__(self, n):
r"""
INPUTS:
- ``n``: a non negative integer
Returns ``self`` to the `n^{th}` power.
EXAMPLES::
sage: S = Monoids().example()
sage: x = S("aa")
sage: x^0, x^1, x^2, x^3, x^4, x^5
('', 'aa', 'aaaa', 'aaaaaa', 'aaaaaaaa', 'aaaaaaaaaa')
"""
if not n: # FIXME: why do we need to do that?
return self.parent().one()
return generic_power(self, n, self.parent().one())
def __pow__(self, n, dummy=None):
if not self.is_endomorphism():
raise TypeError, "self must be an endomorphism."
# todo -- what about the case n=0 -- need to specify the identity map somehow.
return generic_power(self, n)