YAP (was Re: Proposal: Remove Show and Eq superclasses of Num)
Daniel Fischer
daniel.is.fischer at googlemail.com
Thu Nov 3 21:14:23 CET 2011
On Thursday 03 November 2011, 20:01:25, Balazs Komuves wrote:
> It seems to me that a typical Euclidean domain does not have any kind of
> meaningful canonical associate / unit map.
I agree.
> 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.
> - 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.
>
> (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'.
> "unit" is already a very overloaded word)
Yes.
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