Embedding higher kinded types (monads!) into the untyped lambda calculus

Question

It\'s possible to encode various types in the untyped lambda calculus through higher order functions.

Examples:
zero  = λfx.      x
one   = λfx.     fx
two   = λ         


        

Answer 1

Well, we already have tuples and booleans, hence we can represent Either and in turn any non-recursive sum type based on that:

type Either a b = (Bool, (a, b))
type Maybe a    = Either () a

And Maybe is a member of the Monad type class. Translation to lambda notation is left as exercise.