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

[Martin Rubey]

Dear all,

I will first describe my problem with two examples occurring in practice, and
finally try to give a more abstract description. The question is, how to
structure categories in these cases.

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)

Since "graphic matroids" are so important, I would like to have a category
"GraphicMatroidCat", and since they are matroids I'd like to have it inherit
from "MatroidCat". But this is impossible, since the class is not closed under
duality.

Example 2, Differential Equations

A Holonomic function is a function f -- say in one variable x -- that satisfies
a differential equation with polynomial coefficients:

  p_k(x) f^(k) (x) + ... + p_2(x) f''(x) + p_1(x) f'(x) + p_0(x) f(x) = q(x)

where f^(k) is the k_th derivative of f and p_i and q are polynomials.

It is quite easy to see that the class of holonomic functions is closed under
integration, i.e., integrating a holonomic function yields again a holonomic
function. This is quite an important property, of course, so a priori, I'd like
to export it in "HolonomicCat".

If we take k=0 in the equation above we obtain the class of rational
functions. Since rational functions are so important, I'd like to have a
category "RatCat". However, we all know that the class of rational functions is
*not* closed under integration, the simplest example being 1/x.

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.

Can you propose a "natural" hierarchy of categories for this situation?  Since
this occurs so often, I hope that there is a nice solution...

Martin