Re: [Axiom-developer] Curiosities with Axiom mathematical structures

[Ralf Hemmecke]

> Do you mean that passing operations to the categories would be a feasible
> approach? I.e., being allowed to say something like
>
> Monoid(m:(%,%)->%): Category == with
>        square: % -> %
>
>      add
>        square a == m(a,a)

Martin, stop writing in SPAD. I thought you also agreed that new things 
have to be written in Aldor. And there you would to write the default 
implementation with "default" and not with "add". Otherwise there would 
be an anonymous domain "add {square ...}" inside a with clause.

> Of course, one would have to rethink this several times, the suggested notation
> here is clearly not yet perfect. One might object that this notation violates
> the possibility of creating anonymous categories, but I suspect that this
> wouldn't make sense anyway.

Yes, one has to rethink that quite a lot. The problem with your 
suggestion is the following

1) You cannot simply ask

  Integer has Monoid

because you have to give a parameter.

2) If you say something like

MyMonoid(T: Type, m: (T, T) -> T): Category == with {
  square: T-> T;
  default {square(t: T): T == m(t, t)}
}

then it is perfect Aldor.
Then, however, you ask

  Integer has MyMonoid(Integer, *)

and it will return true only if you have said

extend Integer: MyMonoid(Integer, *) == add;

somewhere.
(But probably you had something else in mind.)

Although I don't really like that an AbelianMonoid is not a Monoid, and 
although I think that renaming during inheritance would be nearer to 
mathematics... after all the discussion here, I somehow think that the 
design in Axiom is not really bad. The reason is that I have not seen a 
clear case where renaming would be over-advantageous.

I'd like to say

  Integer has Monoid

instead of

  Integer has Monoid(*, 1);
  Integer has Monoid(+, 0);

Simply think of a category Foo with hundreds of exported function, would 
you like to write

  Dom has Foo(f1, f2, ..., f100)

??? That is not really handy.

Maybe we should open up a page on AxiomWiki and list some cases where 
one could really benefit from renaming. I guess the list would be quite 
short and not comparable with the effort such an Aldor extension would cost.

Ralf