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

[Ralf Hemmecke]

On 04/04/2006 10:11 AM, Martin Rubey wrote:
> Example 1, Matroids

> A "matroid" is a mathematical structure with one very, very important
> operation, namely "dualizing" which transforms a given matroid into
> another. Thus, one is tempted to have a category "MatroidCat", which exports an
> operation "dual: % -> %".

> However, a very important class of matroids, called "graphic matroids", do have
> this operation only if the matroid is "planar". (In fact, "graphic matroids"
> are simply undirected, unweighted graphs)

From your description it is totally obvious: "graphic matroids" are not 
"matroids".

Since I cannot believe this, I would simply say, your "dual" function is 
not as inherent to a matroid as you think. You probably would like to create

define MatroidCategory: Category == with {
  -- don't know yet
}

define DualizableMatroidCategory: Category == with {
  MatroidCategory;
  dual: % ->%;
}

define GraphicalMatroidCategory: Category == with {
  MatroidCategory;
  if ... --express "planar" here
  then DualizableMatroidCategory;
}


> The general case

> 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.

So think of Z (integers) and N (natural numbers) and the operation
  -: % -> %
You would probably never say that N is an additive Group where the 
negation fails. Well, you could do this, but that is not mathematically 
natural. You simply say N is not an additive group.

Well, of course, you don't declare N to be of AbelianGroup.

I see your point, but at the moment I cannot think of a good advice for 
the general case. For the "holonomic" example, it sounds a bit strange 
to say that rational functions do not inherit from holonomic functions.
But what is true is that rational function don't inherit the closure 
properties. So maybe as above that are two categories and one of them is 
the "ClosurePropertyCategory".

I don't yet know whether this is the best thing, but I cannot think of 
anything better at the moment.

Ralf