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

[Christian Aistleitner]

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.

> 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.

> Example 2, Differential Equations

Again the same problem as above.

Of course you can introduce a category

OpSubstructureType: Category == with {
  opable?: % -> Boolean;
  +++if   opable? a, then op( a ) dualizes.
  +++if ~ opable? a, then the behaviour of op is undefined
  op: % -> %; -- op( a ) with ~ opable?( a ) crashes the program
  opIfCan: % -> Partial %
}

and use it in all places, but think about what is the more abstract  
setting and what the specialized one.
For me, a property of a generalization has to hold in its spezialization.
And a property of a specialization does not have to a generalization.

So I'd suggest

+++\begin{addescription}{provides a category for graphical matroids}
+++Some element of a \adthistype may be dualizable.
+++\end{addescription}
GraphicalMatroidType: Category == with {

  dualizeable?: % -> Boolean;

  +++\begin{addescription}{computes the dual element}
  +++if   dualizeable? a, then op( a ) dualizes.
  +++if ~ dualizeable? a, then the behaviour of op is undefined
  +++\end{addescription}
  dualize: % -> %;

  dualizeIfCan: % -> Partial %
}

and

+++\begin{addescription}{provides a category for matroids}
+++Every element of a \adthistype has to be dualizable.
+++\end{addescription}
MatroidType: Category == with {
  GraphicalMatroidType;
  default {
    dualizeable?( a: % ): Boolean == {
      true
    };
    dualizeIfCan( a: % ): Partial % == {
      [ (address@hidden) ]
    };
  }
}


Kind regards,
Christian