class Monad m where
(>>=) :: m a -> ( a -> m b) -> m b
(>>) :: m a -> m b -> m b
return :: a -> m a
and these functions must obey the properties
return a >>= k = k a
m >>= return = m
m >>= (\x -> k x >>= h) = (m >>= k) >>= h
Anything other than the definition I gave is an analogy that will only work for certain examples, that is, unless you can mathematically prove that your analogy is isomorphic to this definition
How does your "definition" (I will take that to mean "an abstract wrapper for some properties that you can apply sequential transforms on" from your first comment) apply to e.g. the monad whose algebras are monoids?