def _mul_(self, other):
"""
Multiply ``self`` by ``other``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a*b^2*e*d
F[0]*F[1]^2*F[3]*F[4]
"""
return self.__class__(self.parent(), dict_addition([self._monomial, other._monomial]))
def _mul_(self, other):
"""
Multiply ``self`` by ``other``.
EXAMPLES::
sage: F = FreeAbelianMonoid(index_set=ZZ)
sage: a,b,c,d,e = [F.gen(i) for i in range(5)]
sage: a*b^2*e*d
F[0]*F[1]^2*F[3]*F[4]
"""
return self.__class__(self.parent(),
dict_addition([self._monomial, other._monomial]))
def _mul_(self, other):
"""
Multiply ``self`` by ``other``.
EXAMPLES::
sage: G = Groups().Commutative().free(index_set=ZZ)
sage: a,b,c,d,e = [G.gen(i) for i in range(5)]
sage: a*b^2*e^-1*d
F[0]*F[1]^2*F[3]*F[4]^-1
sage: (a*b^2*d^2) * (d^-4*b^-2*e)
F[0]*F[3]^-2*F[4]
sage: (a*b^-2*d^2) * (d^-2*b^2*a^-1)
1
"""
return self.__class__(self.parent(),
dict_addition([self._monomial, other._monomial]))
def _mul_(self, other):
"""
Multiply ``self`` by ``other``.
EXAMPLES::
sage: G = Groups().Commutative().free(index_set=ZZ)
sage: a,b,c,d,e = [G.gen(i) for i in range(5)]
sage: a*b^2*e^-1*d
F[0]*F[1]^2*F[3]*F[4]^-1
sage: (a*b^2*d^2) * (d^-4*b^-2*e)
F[0]*F[3]^-2*F[4]
sage: (a*b^-2*d^2) * (d^-2*b^2*a^-1)
1
"""
return self.__class__(
self.parent(), dict_addition([self._monomial,
other._monomial]))
def _add_(self, other):
"""
Return ``self`` added to ``other``.
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: (dx*dy) + dz + x^3 - 2
dx*dy + dz + x^3 - 2
"""
F = self.parent()
return self.__class__(F, dict_addition([self.__monomials, other.__monomials]))
d = copy(self.__monomials)
zero = self.parent().base_ring().zero()
for m,c in other.__monomials.iteritems():
d[m] = d.get(m, zero) + c
if d[m] == zero:
del d[m]
return self.__class__(self.parent(), d)
