On 2 Apr 2008, at 16:20, Loup Vaillant wrote:
class AdditiveSemiMonoid a where (+) :: a -> a -> a
Err, why *semi* monoid? Plain "monoid" would not be accurate?
I found an example where it is crucial that the monoid has a unit: When given a monoid m, then one can also define an m-algebra, by giving a structure map (works in 'hugs -98'): class Monad m => MAlgebra m a where smap :: m a -> a Now, the set of lists on a set A is just the free monoid with base A; the list monad identifies the free monoids, i.e., the lists. So this then gives Haskell interpretation class Monoid a where unit :: a (***) :: a -> a -> a instance Monoid a => MAlgebra [] a where smap [] = unit smap (x:xs) = x *** smap xs Here, I use (***) to not clash with the Prelude (*). But the function "product" here shows up as the structure map of a multiplicative monoid. Similarly, "sum" is the structure map of the additive monoids. And (++) is just the monoid multiplication of the free monoids. Hans