Joachim Durchholz wrote:
Nope, monoid is a special case of monad (the case where all input and output types are the same). (BTW monoid is associativity + neutral element. Not 100% sure whether monad's "return" qualifies as a neutral element, and my monoid-equals-monotyped-monad claim above may fall down if it is not.
For every monad M, the type M () is a monoid, with mempty = return () x <> y = x >> y Does every monoid arise this way? Yes: Since base-4.9 there is the monad instance Monoid a => Monad ((,) a) and of course a and (a,()) are isomorphic (disregarding bottoms). Also, there is the famous tongue-in-cheek saying "monads are just monoids in the category of endofunctors" which is explained in numerous blog posts and Youtube videos. The monoid M () above has funny properties. Specialize M = [] and obtain the monoid of natural numbers under multiplication. Since (>>) has the more general type Monad m => m a -> m b -> m b We have a monoid action of M () on any type M b. In case of lists, it multiplies elements: [(),()] >> xs doubles the number of every element of the list xs. Olaf