Monoids for the option of a function

gistfile1.scala

import scalaz._
import Scalaz._

trait A {
  def increment: A
}
case class A1(i: Int) extends A {
  def increment = A1(i+20)
}

val f0: Option[A => A]  = None
val f1: Option[A => A] = Some((a: A) => a.increment)

val g0: Option[Int => Int]  = None
val g1: Option[Int => Int] = Some((a: Int) => a + 20)


implicit val m = new Monoid[A => A] {
  def zero = identity
  def append(f1: A => A, f2: =>(A => A)) = f1 andThen f2
}

// m kicks in
(~f0).apply(A1(3))  // A1(3)
(~f1).apply(A1(3))  // A1(23)

// the monoid kicking in is the Monoid[Option[T]] where T is a Monoid
// in the case of Int => Int, it's the Monoid created from a function returning a type B where B 
// has a Monoid (Int in this case
(~g0).apply(3)      // 0
(~g1).apply(3)      // 23

note that If you change the types of f0,f1,g0,g1 From Option[Int => Int] to Option[Endo[Int]] then you get the same behavior as your 'm' monoid