# Scala Monads

## Examples

#### Monad Definition

Informally, a monad is a container of elements, notated as F[_], packed with 2 functions: flatMap (to transform this container) and unit (to create this container).

Common library examples include List[T], Set[T] and Option[T].

#### Formal definition

Monad M is a parametric type M[T] with two operations flatMap and unit, such as:

```
trait M[T] {
def flatMap[U](f: T => M[U]): M[U]
}
def unit[T](x: T): M[T]
```

These functions must satisfy three laws:

1.Associativity: (m flatMap f) flatMap g = m flatMap (x => f(x) flatMap g) That is, if the sequence is unchanged you may apply the terms in any order. Thus, applying m to f, and then applying the result to g will yield the same result as applying f to g, and then applying m to that result.

2.Left unit: unit(x) flatMap f == f(x) That is, the unit monad of x flat-mapped across f is equivalent to applying f to x.

3.Right unit: m flatMap unit == m This is an 'identity': any monad flat-mapped against unit will return a monad equivalent to

itself.

#### Example:

```
val m = List(1, 2, 3)
def unit(x: Int): List[Int] = List(x)
def f(x: Int): List[Int] = List(x * x)
def g(x: Int): List[Int] = List(x * x * x)
val x = 1
```

1. Associativity:

```
(m flatMap f).flatMap(g) == m.flatMap(x => f(x) flatMap g) //Boolean = true
//Left side:
List(1, 4, 9).flatMap(g) // List(1, 64, 729)
//Right side:
m.flatMap(x => (x * x) * (x * x) * (x * x)) //List(1, 64, 729)
```

2. Left unit

```
unit(x).flatMap(x => f(x)) == f(x)
List(1).flatMap(x => x * x) == 1 * 1
```

3. Right unit

```
//m flatMap unit == m
m.flatMap(unit) == m
List(1, 2, 3).flatMap(x => List(x)) == List(1,2,3) //Boolean = true
```

#### Standard Collections are Monads

Most of the standard collections are monads (List[T], Option[T]), or monad-like (Either[T], Future[T]). These collections can be easily combined together within for comprehensions (which are an equivalent way of writing flatMap transformations):

```
val a = List(1, 2, 3)
val b = List(3, 4, 5)
for {
i <- a
j <- b
} yield(i * j)
```

The above is equivalent to:

```
a flatMap {
i => b map {
j => i * j
}
}
```

Because a monad preserves the data structure and only acts on the elements within that structure, we can endless chain monadic datastructures, as shown here in a for-comprehension.