On 28/12/2012, Herbert Valerio Riedel
I just realized what confuses me about this bounded maximum/minimum definitions: In the math literature I've been exposed to so far (see also [1][2]), to satisify the definition of a maximal/minimal element of a set that said element has to be actually contained in that set.
So it seems very confusing to me to call the element resulting from
(Ord a, Bounded a) => Monoid (Max a)
with 'Max a' isomorphic to 'a' a proper "maximum" (as it violates the definition for 'mempty'); on the other hand, the term "supremum"[3] seems to match the semantics of the Monoid above better.
[1]: http://en.wikipedia.org/wiki/Maximal_element#Definition [2]: http://mathworld.wolfram.com/Maximum.html [3]: http://en.wikipedia.org/wiki/Supremum
Ah. Yes, thanks for the clarification, to me at least. So it seems that our type of "maximum" and "minimum" is wrong; rather, it ought to be this: maximum, minimum :: (Ord a, Foldable v) => v a -> Maybe a and (Max a) and (Min a) are not properly monoids. We could define these: Supr, Infi :: * -> * instance (Ord a, Bounded a) => Monoid (Supr a) instance (Ord a, Bounded a) => Monoid (Infi a) but I'm not sure whether we ought to. Cheers, Strake