Tyson Whitehead writes:
I was also wondering about the EuclidanDomain "associate" and "unit" functions. YAP requires, for all x
x = associate x * unit x (where unit x has an multiplicative inverse) associate (unit x) = unit (associate x) = 1
http://hackage.haskell.org/packages/archive/yap/0.1/doc/html/Data-YAP- Algebra.html#t:EuclideanDomain
This seem very much like generalized absolute and signum functions.
The two concepts are similar for Int and Integer, except that signum 0 = 0 while unit 0 = 1, but the meanings of abs and signum on other types generalize in a different way. These functions are pretty common in computer algebra systems.
The wikipedia page on Euclidean Domains, however, doesn't mention such functions. Rather it works with the concept of a "euclidian" function which gives a mapping onto the non-negative integers such that
euclidean (mod a b) < euclidean b whenever mod a b /= 0
http://en.wikipedia.org/wiki/Euclidean_domain#Definition
Are these two viewpoints connected? Does a Euclidian domain's euclidian function somehow give rise to the "associate" and "unit" functions?
The purpose of the EuclideanDomain class is to allow us to use Euclid's algorithm, defining gcd :: EuclideanDomain a => a -> a -> a and with that instance EuclideanDomain a => Field (Ratio a) The Euclidean function is the measure function into a well-ordered set (the naturals) used in the proof that gcd terminates. You don't actually use it in the code. On the other hand you do need to be able to get a canonical associate.