[Axiom-developer] Re: [Aldor-l] operations working in general, but not in special cases -- help needed

[Martin Rubey]

"Christian Aistleitner" <address@hidden> writes:

> Dear Martin,
> 
> > We have a category A with an operation op: % -> %. However, there are  
> > natural
> > subdomains of domains of A, which are no longer closed under op.
> 
> if I understood you correctly, you did not mean "closed under op" but "op  is 
> a
> partial function" -- which is a completely different thing.

Could you please elaborate. Ideally, the function "dual" would not be available
for graphical matroids and I'd have yet another class
"PlanarGraphicalMatroids".

Or, maybe even better, the dual of a graphical non-planar matroid would simply
yield a general matroid.

Same thing with differential equations: integrating a rational function should
either be unavailable or yield a holonomic function.

In case the function is unavailable, the axiom interpreter would automatically
try to coerce it into a domain where the operation is indeed valid. So there is
no problem for users.

> > Example 1, Matroids
> 
> > A "matroid" is a mathematical structure with one very, very important
> > operation, namely "dualizing" ...
> 
> If a matroid is required to have a "dualizing function", then a graphical
> matroid cannot be a matroid. Simply because it does not provide the total
> function "dualize"
> 
> However, a graphical matroid can "contain" a matroid.
> 
> So the abstract setting is "graphical matroid" and the specialized one is
> "matroid". Not the other way round.

In a way you are right, of course - I followed the same reasoning
originally. However, I do feel a little uneasy saying that a graphical matroid
is not a matroid...

Furthermore, there is another complication:

For a certain class of graphical matroids, namely those which are 3-connected,
we can recover the vertices. I.e.,

general Matroids have duals but no vertices

graphic non-planar three-connected Matroids do not have duals but do have
vertices

So, what to do?

Martin