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

[Bertfried Fauser]

On 4 Apr 2006, Martin Rubey wrote:

Hi Martin,

a quite interesting question...

> 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...

In general, a mathematical substructure inherites properties of its
ancestor if and only if the process of producing the substructure is
functorial (a natural mapping). There is the general construction of free
objects, eg free algebra, free group, free monoid etc and the theorem that
every such substructure is an image of teh free structure.
        However, if a free object comes with additional features these
features might not be transported in a meaningfull manner into a
subcategory. To add a further example, take a graded algebra and divide
out a ungraded ideal, then the result is an ungraded alegbra (Tensor
algebra -> Lie algebra), while eg the even od grading might survife in
this setting (-> super Lie alegbra).

Possible solution:
        If I understand the inhertitance correctly, it might be seen as a
(not necessarily) natural map from a (free) category to a subcategory
specified by having additional features (obtained by the mapping). Hence
it might be necessary to enlarge the structure of the definition of a
category/domain:

Currently we have a descriptiv part and an implementation part, there
could be a third part in the definition with prevents certain functions
from the parent category to be inherited or (if possible) alter them to
check if they are applicable.


Category Matroid

  expoerts dual

then

Category GraphMatroid

  inherits Matroid
=>modifies dual -> dual= if has planar then dual else error
  exports  new functionality

the second line is new and would need an alteration of spad and aldor
compilers. However, such an copntrolled overloading should not be to
difficult to be implemented and could help to manage the cases you brought
up, as other ones, eg the change of gradings etc.

In general I doubt that mathematics has a plain tree structure so that
there is a root category. Even is Set (Bourbaki tradition, but see
Lawvere!) is used to do so, nobody would start with Set to do say
differential topology. Also a computer algebra system which would tried to
do this would be doomed to fail.

ciao
BF.

% PD Dr Bertfried Fauser
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<http://www.mis.mpg.de>
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