Hi,
> > So, if you want a general framework, I could imagine that the following
> > could work. Provide Parent with a method, say,
> > def to_generator_data(self,x)
> > which by default raises a NotImplementedError.
> > And then, there could be a method
> > def from_generator_data(self, data, gens=None)
> > that constructs an element out of the data returned by to_generator_data,
> > but with an optional argument `gens` that defaults to the list of
> > generators.
> >
> > The specification is:
> > self.from_generator_data(self.to_generator_data(x))==x
> > for all x in self. Such a method could be useful anyway, because it
> > provides copying and pickling of elements.
> >
> > And if R, S are parents and the generators of R are
> > mapped to a list gens_images, then
> > S.from_generator_data(R.from_generator_data(x), gens_images)
> > should return the image of an element x of R under a morphism from R to
> > S given by gens_images.
>
> Yes, I that sounds like a reasonable proposal. However, there's still
> the question as to which category you're considering the generators
> over--we'd probably have to make that explicit.
>
> > Only problem: Someone needs to implement it. There could be general
> > implementations on the category level, though.
Yes ! This is something which is at least planned and some Category are
starting to support such features.
There are at least
S.semigroup_generators for a semi group (no inverse)
S.group_generators for a group
A.algebra_generators for an algebra
All these methods should return a Family (G_i)_{i in I} which are some king of
Lazy/Intelligent dictionaries.
One solution which is not performance-wise optimal but mathematically sound is
to have to_generator_data actually return an element of the
free(Group/Algebra/Whatever) generated by the index-set I. Then building a
generic morphism form the FreeWhatever to A is easy to implement in a generic
way. The extra cost comes from building an actual Element of the FreeWhatever
compared to the underlying data-structure.
I've implemented something like that in Operads on Sage-Combinat queue.
Cheers,
Florent
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