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.