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# Scala example source code file (Monoid.scala)

This example Scala source code file (Monoid.scala) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Scala by Example" TM.

## Java - Scala tags/keywords

applicative, applicativemonoid, equal, monoid, plusempty

## The Monoid.scala Scala example source code

```package scalaz

////
/**
* Provides an identity element (`zero`) to the binary `append`
* operation in [[scalaz.Semigroup]], subject to the monoid laws.
*
* Example instances:
*  - `Monoid[Int]`: `zero` and `append` are `0` and `Int#+` respectively
*  - `Monoid[List[A]]`: `zero` and `append` are `Nil` and `List#++` respectively
*
* References:
*  - [[http://mathworld.wolfram.com/Monoid.html]]
*
* @see [[scalaz.syntax.MonoidOps]]
* @see [[scalaz.Monoid.MonoidLaw]]
*
*/
////
trait Monoid[F] extends Semigroup[F] { self =>
////
/** The identity element for `append`. */
def zero: F

// derived functions
/**
* For `n = 0`, `zero`
* For `n = 1`, `append(zero, value)`
* For `n = 2`, `append(append(zero, value), value)`
*/
def multiply(value: F, n: Int): F =
if (n <= 0) zero else multiply1(value, n - 1)

/** Whether `a` == `zero`. */
def isMZero(a: F)(implicit eq: Equal[F]): Boolean =
eq.equal(a, zero)

final def ifEmpty[B](a: F)(t: => B)(f: => B)(implicit eq: Equal[F]): B =
if (isMZero(a)) { t } else { f }

final def onNotEmpty[B](a: F)(v: => B)(implicit eq: Equal[F], mb: Monoid[B]): B =
ifEmpty(a)(mb.zero)(v)

final def onEmpty[A,B](a: F)(v: => B)(implicit eq: Equal[F], mb: Monoid[B]): B =
ifEmpty(a)(v)(mb.zero)

/** Every `Monoid` gives rise to a [[scalaz.Category]], for which
* the type parameters are phantoms.
*
* @note `category.monoid` = `this`
*/
final def category: Category[λ[(α, β) => F]] =
new Category[λ[(α, β) => F]] with SemigroupCompose {
def id[A] = zero
}

/**
* A monoidal applicative functor, that implements `point` and `ap`
* with the operations `zero` and `append` respectively.  Note that
* the type parameter `α` in `Applicative[λ[α => F]]` is
* discarded; it is a phantom type.  As such, the functor cannot
* support [[scalaz.Bind]].
*/
final def applicative: Applicative[λ[α =>F]] =
new Applicative[λ[α => F]] with SemigroupApply {
def point[A](a: => A) = zero
}

/**
* Monoid instances must satisfy [[scalaz.Semigroup.SemigroupLaw]] and 2 additional laws:
*
*  - '''left identity''': `forall a. append(zero, a) == a`
*  - '''right identity''' : `forall a. append(a, zero) == a`
*/
trait MonoidLaw extends SemigroupLaw {
def leftIdentity(a: F)(implicit F: Equal[F]) = F.equal(a, append(zero, a))
def rightIdentity(a: F)(implicit F: Equal[F]) = F.equal(a, append(a, zero))
}
def monoidLaw = new MonoidLaw {}

////
val monoidSyntax = new scalaz.syntax.MonoidSyntax[F] { def F = Monoid.this }
}

object Monoid {
@inline def apply[F](implicit F: Monoid[F]): Monoid[F] = F

////

/** Make an append and zero into an instance. */
def instance[A](f: (A, => A) => A, z: A): Monoid[A] =
new Monoid[A] {
def zero = z
def append(f1: A, f2: => A): A = f(f1,f2)
}

private trait ApplicativeMonoid[F[_], M] extends Monoid[F[M]] with Semigroup.ApplySemigroup[F, M] {
implicit def F: Applicative[F]
implicit def M: Monoid[M]
val zero = F.point(M.zero)
}

/**A monoid for sequencing Applicative effects. */
def liftMonoid[F[_], M](implicit F0: Applicative[F], M0: Monoid[M]): Monoid[F[M]] =
new ApplicativeMonoid[F, M] {
implicit def F: Applicative[F] = F0
implicit def M: Monoid[M] = M0
}

def liftPlusEmpty[A](implicit M0: Monoid[A]): PlusEmpty[λ[α => A]] =
new PlusEmpty[λ[α => A]] {
type A0[α] = A
def empty[A]: A0[A] = M0.zero
def plus[A](f1: A0[A], f2: => A0[A]): A0[A] = M0.append(f1, f2)
}

/** Monoid is an invariant functor. */
implicit val monoidInvariantFunctor: InvariantFunctor[Monoid] =
new InvariantFunctor[Monoid] {
def xmap[A, B](ma: Monoid[A], f: A => B, g: B => A): Monoid[B] = new Monoid[B] {
def zero: B = f(ma.zero)
def append(x: B, y: => B): B = f(ma.append(g(x), g(y)))
}
}

////
}
```

## Other Scala examples (source code examples)

Here is a short list of links related to this Scala Monoid.scala source code file:

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