Re: [Axiom-developer] A curious algebra failure

[Bill Page]

On 8/12/07, Gabriel Dos Reis <address@hidden> wrote:
> On Sat, 11 Aug 2007, Bill Page wrote:
> | ...
> | Unnamed categories are just values of type Category.
>
> But, that is not what Axiom Book says, section 12.12, page 525:
>
>    The part of a category to the right of a with is also regarded as
>    a category -- an "anonymous category". [...]
>    [ rewriting packages with named categories skipped ]
>    There is no reason, however, to give this list of exports a name since no
>    other domain of package exports it.  In fact, it is rare for a package to
>    export a named category.  [...]
>
> This, to me, suggest that the semantics described and intended in the
> Axiom Book is that two unnamed categories always yields different categories
> irrespective of their bodies.
>

You must be reading between the lines :-) since I certainly can not
deduce that two unnamed categories are always distinct from the
quotation. In fact the opposite. The discussion of exporting a named
category here is rather gratuitous unless you assume that there is a
better reason for exporting "anonymous" categories than simple
convenience. The reason why most packages export anonymous categories
is precisely because of how named categories are interpreted:
categories with different names but identical structure are considered
distinct. This is important because Axiom wants to associate these
*names* with mathematical properties. In many cases the name must
carry the proper mathematical interpretation irrespective of whether
or not the categories have the same structure.

See the examples at:

http://wiki.axiom-developer.org/SandBoxCategories

But notice that these examples also illustrate what I think is an
error in the result of evaluating has for category-valued expressions
in Spad.

> That may not be what we want or would like, but I believe that is what was
> intended in the Axiom Book.
>

No can't be true, otherwise how would any of the Axiom library
actually work? (As Stephen has already pointed out.)

>
> | They are distinct only if the values are distinct. 'with' is
> | used to form a category-valued expression.
>
> I don't get this.  Please, could you give more Spad examples of what you mean?
>

See

http://wiki.axiom-developer.org/SandBoxCategories

Regards,
Bill Page