Monoid extends the Semigroup type class, adding an empty method to semigroup's combine. The empty method must return a value that when combined with any other instance of that type returns the other instance, i.e.

(combine(x, empty) == combine(empty, x) == x)

For example, if we have a Monoid[String] with combine defined as string concatenation, then empty = "".

Having an empty defined allows us to combine all the elements of some potentially empty collection of T for which a Monoid[T] is defined and return a T, rather than an Option[T] as we have a sensible default to fall back to.

First some imports.

import cats._
import cats.implicits._

And let's see the implicit instance of Monoid[String] in action.

import cats.implicits._

Monoid[String].empty should be(res0)
Monoid[String].combineAll(List("a", "b", "c")) should be(res1)
Monoid[String].combineAll(List()) should be(res2)

The advantage of using these type class provided methods, rather than the specific ones for each type, is that we can compose monoids to allow us to operate on more complex types, e.g.

Monoid[Map[String, Int]].combineAll(List(Map("a" → 1, "b" → 2), Map("a" → 3))) should be(res0)
Monoid[Map[String, Int]].combineAll(List()) should be(res1)

This is also true if we define our own instances. As an example, let's use Foldable's foldMap, which maps over values accumulating the results, using the available Monoid for the type mapped onto.

val l = List(1, 2, 3, 4, 5)
l.foldMap(identity) should be(res0)
l.foldMap(i ⇒ i.toString) should be(res1)

To use this with a function that produces a tuple, we can define a Monoid for a tuple that will be valid for any tuple where the types it contains also have a Monoid available. Note that cats already defines it for you.

implicit def monoidTuple[A: Monoid, B: Monoid]: Monoid[(A, B)] =
  new Monoid[(A, B)] {
    def combine(x: (A, B), y: (A, B)): (A, B) = {
      val (xa, xb) = x
      val (ya, yb) = y
      (Monoid[A].combine(xa, ya), Monoid[B].combine(xb, yb))
    def empty: (A, B) = (Monoid[A].empty, Monoid[B].empty)

This way we are able to combine both values in one pass, hurrah!

val l = List(1, 2, 3, 4, 5)
l.foldMap(i ⇒ (i, i.toString)) should be(res0)