E.g. Arrays are monads under concatenation. So why people then say easily "Option type is monad" without stating under which operation? I assume Option is monad under "all" and "any" function, but not sure.
1. A type constructor M of a single argument 2. A function of type a -> M a 3. A function of type M (M a) -> M a
subject to the monad laws. So the 'array monad' is the triple (Array, \x -> [x], Array.concat) and the 'option monad' is the triple (Option, \x -> Some(x), \x -> match x | Some(o) => o; None => None). Saying a type 'is' a monad is imprecise but just means it has an associated monad instance. I've never seen a type with multiple monad instances but I don't know if there could be one.
There can, it's just like there can two monoid structures on the same set. In fact if you have an example of a set S which has two monoid structures, you have an example of a functor with two monad structures: _ × S. join multiplies the two elements of S, so you get a different monad structure for every monoid structure on S.
Btw, I think you got a mistake there, in the third part of Option should be "Some(Some(o)) => Some(o), Some(None) => None, None => None", otherwise o can be not an Option at all.
let join_option oo =
match oo with
| Some(o) -> o
| None -> None [] => []
[[]] => []
[[a]] => [a]