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

From: Martin Rubey
Subject: Re: [Axiom-developer] Curiosities with Axiom mathematical structures
Date: 09 Mar 2006 15:46:14 +0100
User-agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.4

Ralf Hemmecke <address@hidden> writes:

> > 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.

Well, I'm not yet fluent in Aldor. I'm waiting for your talk at the Workshop...

> > 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.

I wouldn't want to ask "Integer has Monoid", since this doesn't make any sense
to me. I'd like to ask "Integer has Monoid(Integer, *)" or 
"Integer has Monoid(*)"

> 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.)

So, in fact in Aldor we can solve the "problem"?

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

Word: BasicType == add {
   add {*(a: %, b: %):%  == concat(a,b) }

extend Word: Monoid(Word, *) == add;

??? If this is the case, I know what I want!!!

> 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)

no, but wait a moment: It is obvious to me that I don't want to have all
exported functions as parameters. Only certain "defining" functions, like:

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

Can you think of an example where more than, say 5, parameters would be 


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