This is not really a definition of a Monad. For example, there's no mention of the Monad laws. Monads are a very general and powerful abstraction that are not

This is not really a definition of a Monad. For example, there's no mention of the Monad laws. Monads are a very general and powerful abstraction that are not adequately described by your example. My advice to anyone is to read Phil Wadler's seminal paper, it is very easy to read.