There Haskell people go again =) You try to define something in simple normal terms, and off they go adding highly specific and technically correct words to what was supposed to be a simple explanation.

(I'm not a Haskell expert, I dabble, done CIS194, half of RWH, and I have no idea what you said -- which is a common problem I run into in the Haskell world, lots of super helpful people that have forgotten what its like to not speak their language)

I think using Haskell's type syntax is misleading in this context (and doesn't really imply the required associativity (much less commutativity if you wanted to consider Abelian monoids)).

The operation of a monoid maps from pairs of things to things. So in terms of types:

    <a,a> -> a

Or for some `twin` type constructor:

    twin a -> a

This is a bit more suggestive also in terms of F-algebras, where the operation has the following signature (f is a functor, or mappable container):

    f a -> a