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:

*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.*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`

.*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
```

*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)
```

*Left unit*

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

*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.

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