YAP (was Re: Proposal: Remove Show and Eq superclasses of Num)
Balazs Komuves
bkomuves at gmail.com
Fri Nov 4 00:06:34 CET 2011
On Thu, Nov 3, 2011 at 9:14 PM, Daniel Fischer <
daniel.is.fischer at googlemail.com> wrote:
> > Examples:
> >
> > - The Gaussian integers Z[i] (units are 1,-1,i,-i; what would be the
> > associated element of 5+7i ?)
>
> > - Formal power series K[[x]] over a field (units are every series with
> > nonzero constant coefficients),
>
> This one has a fairly canonical representative for the classes of
> associated series: X^n, where n is the index of the first nonzero
> coefficient.
>
While this may seem "canonical" to a human eye, I would argue that it is
rather ad-hoc.
(for example a change of variable x -> (x+1) will change the notion of
"canonical"?)
>
> > - and probably just about any other interesting structure satisfying the
> > definition.
> >
> > A function "a -> a" in a type class suggests to me a canonical mapping.
> > Thus, I would
> > advocate against putting associate/unit into such a Euclidean domain
> > type class.
>
> True, but I think we'd need such functions to have well-defined "canonical"
> factorisations for example.
>
But if there is no canonical factorization, why do we want to force it?
A normalized factorization in an algebra system is something between
a design choice and an implementation detail, from my viewpoint.
>
> > (Independently of this, I also find the name "unit" a bit confusing for
> > something
> > which would be better called "an associated unit";
>
> Except here, where 'associated' means 'equal up to multiplication with a
> unit'.
>
Actually, this seems to be consistent to me :)
(since two "associated units" differ by a multiplication with a unit)
Balazs
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