fmap is defined as a (higher-order) function that fulfills the following relations (in haskell and the . denotes function concatenation):

fmap id = id

fmap (f . g) = (fmap f) . (fmap g)

the definition in R is then:

fmap <- function(f) sapply(x,f)

for the join function the following has to be valid (also in haskell):

join . fmap join = join . join

this translates to the following code identities in R (read === as "is the same as"):

c( fmap(c)(x)) === c(sapply(x,c)) === c(x) === x

Therefore c and sapply define the list monad in R. Can we go one step further and define some notation like the "do" notation in haskell in R? I think this is already done. It is the foreach package.

foreach(x=a) %do% f(x)

is the exact translation of the corresponding haskell monadic "do" notation (again in haskell):

fmap f x = do r <- x return (f r)

join x = do a <- x a

and in R:

fmap(f)(x) === foreach(r=x) %do% f(r) === sapply(x,f)

join(x) === foreach(a=x) %do% a === c(x)

Now hat does this mean for the several foreach extensions (doMC, doMPI, doRedis, doSNOW)?

They are all implementations of monads in R. So it would be very interesting to port all the other monadic types from other functional programming languages to R.