Re: [Haskell-cafe] monoids and monads

On Mon, Jul 26, 2010 at 5:02 PM, Edward Kmett wrote:

On Mon, Jul 26, 2010 at 11:55 AM, John Lato wrote:

...

Hello,

I was wondering today, is this generally true?

instance (Monad m, Monoid a) => Monoid (m a) where  mempty = return mempty  mappend = liftM2 mappend

There are multiple potential monoids that you may be interested in here.

There is the monoid formed by MonadPlus, there is the monoid formed by wrapping a monad (or applicative) around a monoid, which usually forms part of a right seminearring because of the left-distributive law, there are also potentially other monoids for particular monads.

See the monad module in my monoids package:

http://hackage.haskell.org/packages/archive/monoids/0.2.0.2/doc/html/Data-Mo...

I think your monoids package has grown since I last looked at it. I'll take a look.

...

Any resources for how I could develop a means to reason about this sort of property?

The types are not enough.

What you need is the associativity of Kleisli arrow composition and the two identity laws.

The three monad laws are precisely what you need to form this monoid. There are analogous laws for Applicative that serve the same purpose.

Thanks very much. With this and Henning's hints, I think I can make some progress with this now. John