I am trying to implement free groups in sage. I try to inherit from
ParentWithGens, and seems to work. But when i call the .hom procedure
passing the image of the generators, it fails.
This is the declartion of my class:
class FreeGroup(UniqueRepresentation, group.Group,ParentWithGens):
Element=FreeGroupElement
def __init__(self,gens):
self._gens_str_=tuple(gens)
if len(gens)==1:
strgen='("'+gens[0]+'")'
else:
strgen=str(self._gens_str_).replace("'",'"')
self._gap_repr_=gap('FreeGroup'+strgen)
self._gens_dict_=dict([(gens[i],self._gap_repr_.GeneratorsOfGroup()[i
+1]) for i in range(len(gens)-1)])
group.Group.__init__(self)
self._assign_names(gens)
def ngens(self):
return len(self._gens_str_)
def _repr_(self):
return "Free Group on generators "+str(self._gens_str_)
def gen(self,i):
return self([i+1])
def gens(self):
return tuple([self.gen(i) for i in range(self.ngens())])
def generators(self):
return tuple([self.gen(i) for i in range(self.ngens())])
def one(self):
return self([])
def _an_element_(self):
return self.one()
def gap(self):
return self._gap_repr_
def _element_constructor_(self, *args,**kwds):
if len(args)!=1:
return self.element_class(*args,parent=self,**kwds)
x = args[0]
if x==1:
return self.one()
try:
P = x.parent()
except AttributeError:
return self.element_class(x,parent=self,**kwds)
if hasattr(P,'_freegroup_'):
if P.FreeGroup() is self:
return
self.element_class(x.TietzeList(),parent=self,**kwds)
return self.element_class(x,parent=self,**kwds)
And the element class is:
class FreeGroupElement(Element):
def __init__(self,l=[],parent=None):
if parent is None:
raise ValueError, "The parent must be provided"
if len(l)>0:
if (parent.rank()<max(l)) or (-parent.rank() > min(l)):
raise ValueError, "Generators not in the group"
self._gap_=gap(l).AbstractWordTietzeWord(parent._gap_repr_.GeneratorsOfGroup())
self._TietzeList_=map(Integer,list(self._gap_.TietzeWordAbstractWord()))
Element.__init__(self,parent)
def _repr_(self):
return str(self._gap_)
def _mul_(self,other):
C=self.__class__
return
C(list((self._gap_*other._gap_).TietzeWordAbstractWord()),self.parent())
def __cmp__(self,other):
return cmp(self.TietzeList(),other.TietzeList())
def __div__(self,other):
C=self.__class__
return
C(list((self._gap_*other._gap_.Inverse()).TietzeWordAbstractWord()),self.parent())
def __pow__(self,other):
if not other in IntegerRing():
raise TypeError, "exponent must be an integer"
C=self.__class__
return
C(list((self._gap_.POW(gap(other))).TietzeWordAbstractWord()),self.parent())
def gap(self):
"""
Returns a gap representation of the element
EXAMPLES:
sage: G=FreeGroup(('a','b'))
sage: x=G([1,2,-1,-2])
sage: x
a*b*a^-1*b^-1
sage: xg=x.gap()
sage: xg
a*b*a^-1*b^-1
sage: type(xg)
<class 'sage.interfaces.gap.GapElement'>
"""
return self._gap_
def TietzeList(self):
"""
Returns the Tietze List of the element.
EXAMPLES:
sage: G=FreeGroup(('a','b'))
sage: G.inject_variables()
sage: a.TietzeList()
[1]
sage: x=a^2*b^(-3)*a^(-2)
sage: x.TietzeList()
[1, 1, -2, -2, -2, -1, -1]
"""
return self._TietzeList_
def __invert__(self):
return self.__pow__(-1)
def inverse(self):
"""
Returns the inverse of self
"""
return ~self
Mostly it works fine: you can define the group, and operate with its
elements. But when i try to define a morphism by giving the images of
the generators, it fails:
sage: G=FreeGroup(('a','b','c'))
sage: G.inject_variables()
Defining a, b, c
sage: G.hom([b,b,b])
---------------------------------------------------------------------------
TypeError Traceback (most recent call
last)
/home/mmarco/sage-4.7.2/<ipython console> in <module>()
/home/mmarco/sage-4.7.2/local/lib/python2.6/site-packages/sage/
structure/parent_gens.so in
sage.structure.parent_gens.ParentWithGens.hom (sage/structure/
parent_gens.c:3820)()
/home/mmarco/sage-4.7.2/local/lib/python2.6/site-packages/sage/
structure/parent.so in sage.structure.parent.Parent.hom (sage/
structure/parent.c:9556)()
/home/mmarco/sage-4.7.2/local/lib/python2.6/site-packages/sage/
categories/homset.pyc in __call__(self, x, y, check, on_basis)
431 return self.element_class_set_morphism(self, x)
432
--> 433 raise TypeError, "Unable to coerce x (=%s) to a
morphism in %s"%(x,self)
434
435 @lazy_attribute
TypeError: Unable to coerce x (=[b, b, b]) to a morphism in Set of
Morphisms from Free Group on generators ('a', 'b', 'c') to Free Group
on generators ('a', 'b', 'c') in Category of groups
Which is the right way to define a parent with gens in this case? I
have read in the code that ParentWithGens is deprecated, but i haven't
found which is the correct way to do it.
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