Monoid vs MonadPlus [duplicate]

Question

I am very new to both Monads and Monoids and recently also learned about MonadPlus. From what I see, Monoid and MonadPlus both provide a type with a associative binary operation and an identity. (I'd call this a semigroup in mathematical parlance.) So what is the difference between Monoid and MonadPlus?

Answer 1

A semigroup is a structure equipped with an associative binary operation. A monoid is a semigroup with an identity element for the binary operation.

Monads and semigroups

Every monad has to adhere to the monad laws. For our case, the important one is the associativity law. Expressed using >>=:

(m >>= f) >>= g     ≡   m >>= (\x -> f x >>= g) 

Now let's apply this law to deduce the associativity for >> :: m a -> m b -> m b:

(m >> n) >> p       ≡ (m >>= \_ -> n) >>= \_ -> p                     ≡ m >>= (\x -> (\_ -> n) x >>= \_ -> p)                     ≡ m >>= (\x -> n >>= \_ -> p)                     ≡ m >>= (\x -> n >> p)                     ≡ m >> (n >> p) 

(where we picked x so that it doesn't appear in m, n or p).

If we specialize >> to the type m a -> m a -> m a (substituting b for a), we see that for any type a the operation >> forms a semigroup on m a. Since it's true for any a, we get a class of semigroups indexed by a. However, they are not monoids in general - we don't have an identity element for >>.

MonadPlus and monoids

MonadPlus adds two more operations, mplus and mzero. MonadPlus laws state explicitly that mplus and mzero must form a monoid on m a for an arbitrary a. So again, we get a class of monoids indexed by a.

Note the difference between MonadPlus and Monoid: Monoid says that some single type satisfies the monoidal rules, while MonadPlus says that for all possible a the type m a satisfies the monoidal laws. This is a much stronger condition.

So a MonadPlus instance forms two different algebraic structures: A class of semigroups with >> and a class of monoids with mplus and mzero. (This is not something uncommon, for example the set of natural numbers greater than zero {1,2,...} forms a semigroup with + and a monoid with × and 1.)