`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)
```