Handling failure is probably the most important part of building reliable systems. Monads are a common method of encapsulating failure - for instance, Option[T] returns either Some(t) or None, and None represents the failure. But Monads do not actually have any mechanism for handling failure - in principle, a monad which has failed (whatever failure means) can never be resumed. The same is true for applicative functors.

Nevertheless, recovering from a failure is often desirable. This can sometimes be accomplished via ad-hoc methods on monadic types - e.g., Option.getOrElse or Future.fallbackTo. But these are less than desirable since they are far from generic - if you want to build a class which is generic in it's functor (e.g. class FooM[T](monad: Applicative[T])) you simply cannot recover from a failure.

For this reason, the Plus typeclass was created. Plus encapsulates types which are a semigroup or monoid in addition to being a functor.

MonadPlus in Haskell

In Haskell, the primary usage of this pattern is the MonadPlus typeclass. It introduces a new operation mplus :: m a -> m a -> m a and a new element mzero :: m a which must satisfy the following laws:

mzero `mplus` x == x
x `mplus` mzero == x
(x `mplus` y) `mplus` z == z `mplus` (y `mplus` z)

provided of course that both sides can can be evaluated. (It is not always possible to evaluate both sides due to certain infinite structures.)

Additionally, the MonadPlus typeclass interacts with the Monad typeclass in the following way:

mzero >>= f == mzero
x >> mzero == mzero

Maybe and List

A simple example will illustrate the point of MonadPlus.

Prelude> import Control.Monad
Prelude Control.Monad> Just 5 `mplus` Just 7
Just 5
Prelude Control.Monad> Just 5 `mplus` Nothing
Just 5
Prelude Control.Monad> Nothing `mplus` Just 7
Just 7
Prelude Control.Monad> Nothing `mplus` Nothing

So with MonadPlus, the mplus operation basically means "try the first argument, if it fails, fall back to the second argument".

With List, mplus is simple concatenation:

Prelude Control.Monad> [2,3] `mplus` [7]

Plus in Scalaz

Scalaz takes a slightly more complicated approach. A Plus trait exists, and it merely represents the fact that a Functor is also a semigroup:

trait Plus[F[_]]  { self =>
  def plus[A](a: F[A], b: => F[A]): F[A]

Then the PlusEmpty represents the fact that this functor is also a monoid:

trait PlusEmpty[F[_]] extends Plus[F] { self =>
  def empty[A]: F[A]

A Scalaz implicit provides the <+> method:

Some(5) &lt;+&gt; None == Some(5)
None &lt;+&gt; None == None
List(2,3) &lt;+&gt; List(7) == List(2,3,7)

And similarly, the mzero[F[T]] function provides the zero element:

mzero[Option[Int]] == None
mzero[List[String]] == List()

Note that unlike in Haskell, the monadic structure is not required. You can have a Functor which satisfies Plus or PlusEmpty even if it is not a Monad.

Don't confuse <+> and |+|

For every PlusEmpty instance, Scalaz provides a Monoid instance. But it's important not to confuse the <+> and |+| operators. Using the standard Scalaz implicits, the latter operator will sometimes build nested Monoids:

Some(1) |+| Some(2) == Some(3)
Some(1) &lt;+&gt; Some(2) == Some(1)

So in the case of Option[Int], the <+> operator represents the fallback operation of the outer functor only. In contrast, |+| attempts to make the inner argument into a semigroup as well (if it can). This applies to nested monoids as well:

Some(Some(1)) &lt;+&gt; Some(Some(2)) == Some(Some(1))
Some(Some(1)) |+| Some(Some(2)) == Some(Some(3))
Some(None) &lt;+&gt; Some(Some(2)) == Some(None)
Some(None) |+| Some(Some(2)) == Some(Some(2))

Why it's useful

Using the Plus typeclass allows you to write functions/classes which are generic in the underlying functor.

A common pattern in scala-land is to build a cache which returns an object wrapped in a functor. For example, spray caching returns values wrapped in a Future:

trait SprayCache {
  def get(k: K)(expr: =&gt;Future[V]): Future[V]

(This is an example that is not identical to the Spray cache.)

On other occasions it might be useful to return a Validation or even an Option:

trait OptionalCache {
  def apply(k: K)(expr: =&gt;Option[V]): Option[V]

Now suppose we want to generalize - it's our goal to build a generic CacheF trait which parameterizes the functor. I.e., it will be CacheF[M[_],K,V] and the apply method will return an M[V]. We can, in a very straightforward manner, build such a cache. All you need to implement is the put method and the getFromCache method, and only the latter of which actually returns an M[V]:

trait CacheF[M,K,V] {
  protected implicit val ap: ApplicativePlus[M]
  def put(k: K, v: M[V]): Unit

  protected def getFromCache(k: K): M[V]

  def get(k: K)(expr: =&gt;M[V]): M[V] = getFromCache(k) &lt;+&gt; {
      val result: M[V] = expr

This cache encapsulates the semantics of both the SprayCache and the OptionalCache - you pull an object out of the cache, and if it's a failure (or if the cache fails to return), then you fall back to actually computing the object. But it's important that all code is shared between both - the only code which isn't is the actual implementation details of put and getFromCache.

Math is useful

I've often heard it said that higher math is not particularly useful for programming, at least if you are building CRUD apps. On this point I disagree. Although one can build a CRUD app via basic programming skills and sheer force of will, that doesn't mean knowing a little bit more math is useless. Even when building something as simple as a cache, a little bit of higher math can come in handy and make your code a bit more generic.

P.S. Injera, the library I'm working on uses this pattern in slightly more generality. Go check it out, it might come in handy.

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