What does a "monadic structure" and "element of a structure" precisely mean in the context of arbitrary Monad?

Question

Reading through the documentation of Control.Monad I found such description of mapM:

Map each element of a structure to a monadic action, evaluate these actions from left to right, and collect the results

I'm interested, what does “element of a structure” mean for an arbitrary monad? E.g, State or IO monad? What a structure? I know what mapM function does and interesting more in the semantics of this particular words in the description of mapM. Does this "structure" somehow connected with the notion of "algebraic structure"?

Structure is a set with one or more finitary operations defined on it that satisfies a list of axioms.

If so then what are the elements of this structure (carrier set) as I asked for e.g. IO monad? In general?

Answer 1

"Element of a structure" means exactly nothing for an arbitrary monad, because it doesn't reference the monad. The mapM you're referring to is a member of the Traversable typeclass, and the structure it refers to is the Traversable. A Traversable is a data structure that can be mapped over, such as a list or a tree. "Element of a structure" refers to the list's elements.

Answer 2

The 'structure' here is the instance of the Traversable class (i.e. the type constructor t in the signature of mapM).