Monoid是种最简单的typeclass类型。我们先看看scalaz的Monoid typeclass定义：scalaz/Monoid.scala
1 trait Monoid[F] extends Semigroup[F] { self =>
2 ////
3 /** The identity element for `append`. */
4 def zero: F
5 ...
Monoid trait又继承了Semigroup：scalaz/Semigroup.scala
1 trait Semigroup[F] { self =>
2 ////
3 /**
4 * The binary operation to combine `f1` and `f2`.
5 *
6 * Implementations should not evaluate the by-name parameter `f2` if result
7 * can be determined by `f1`.
8 */
9 def append(f1: F, f2: => F): F
10 ...
所以获取一个类型的Monoid实例需要实现zero和append这两个抽象函数。实际上Monoid typeclass也就是支持了append（|+|）这么一个简单的操作。scalaz为一些标准类型定义了Monoid实例：
1 0 |+| 30 //> res0: Int = 50
2 20.some |+| 30.some //> res1: Option[Int] = Some(50)
3 List(1,2,3) |+| List(4,5,6) //> res2: List[Int] = List(1, 2, 3, 4, 5, 6)
4 Tags.Multiplication(3) |+| Monoid[Int @@ Tags.Multiplication].zero
5 //> res3: scalaz.@@[Int,scalaz.Tags.Multiplication] = 3
6 Tags.Conjunction(true) |+| Tags.Conjunction(false)//> res4: scalaz.@@[Boolean,scalaz.Tags.Conjunction] = false
7 Tags.Disjunction(true) |+| Tags.Disjunction(false)//> res5: scalaz.@@[Boolean,scalaz.Tags.Disjunction] = true
8 Monoid[Boolean @@ Tags.Conjunction].zero //> res6: scalaz.@@[Boolean,scalaz.Tags.Conjunction] = true
9 Monoid[Boolean @@ Tags.Disjunction].zero //> res7: scalaz.@@[Boolean,scalaz.Tags.Disjunction] = false
就这么来看好像没什么值得提的。不过Ordering的Monoid倒是值得研究一下。我们先看看Ordering trait：scalaz/Ordering.scala
1 implicit val orderingInstance: Enum[Ordering] with Show[Ordering] with Monoid[Ordering] = new Enum[Ordering] with Show[Ordering] with Monoid[Ordering] {
2 def order(a1: Ordering, a2: Ordering): Ordering = (a1, a2) match {
3 case (LT, LT) => EQ
4 case (LT, EQ | GT) => LT
5 case (EQ, LT) => GT
6 case (EQ, EQ) => EQ
7 case (EQ, GT) => LT
8 case (GT, LT | EQ) => GT
9 case (GT, GT) => EQ
10 }
11
12 override def shows(f: Ordering) = f.name
13
14 def append(f1: Ordering, f2: => Ordering): Ordering = f1 match {
15 case Ordering.EQ => f2
16 case o => o
17 }
18 ...
这里定义了Ordering的Monoid实例。它的append函数意思是：两个Ordering类型值f1,f2的append操作结果：假如f1是EQ就是f2，否则是f1：
1 (Ordering.EQ: Ordering) |+| (Ordering.GT: Ordering)
2 //> res8: scalaz.Ordering = GT
3 (Ordering.EQ: Ordering) |+| (Ordering.LT: Ordering)
4 //> res9: scalaz.Ordering = LT
5 (Ordering.GT: Ordering) |+| (Ordering.EQ: Ordering)
6 //> res10: scalaz.Ordering = GT
7 (Ordering.LT: Ordering) |+| (Ordering.EQ: Ordering)
8 //> res11: scalaz.Ordering = LT
9 (Ordering.LT: Ordering) |+| (Ordering.GT: Ordering)
10 //> res12: scalaz.Ordering = LT
11 (Ordering.GT: Ordering) |+| (Ordering.LT: Ordering)
12 //> res13: scalaz.Ordering = GT
如果我用以上的特性来比较两个String的长度：如果长度相等则再比较两个String的字符顺序。这个要求刚好符合了Ordering Monoid实例的append操作：
1 3 ?|? 4 //> res14: scalaz.Ordering = LT
2 "abc" ?|? "bac" //> res15: scalaz.Ordering = LT
3 def strlenCompare(lhs: String, rhs: String): Ordering =
4 (lhs.length ?|? rhs.length) |+| (lhs ?|? rhs) //> strlenCompare: (lhs: String, rhs: String)scalaz.Ordering
5
6 strlenCompare("abc","aabc") //> res16: scalaz.Ordering = LT
7 strlenCompare("abd","abc") //> res17: scalaz.Ordering = GT
这个示范倒是挺新鲜的。
好了，单看Monoid操作会觉着没什么特别，好像不值得研究。实际上Monoid的主要用途是在配合可折叠数据结构（Foldable）对结构内部元素进行操作时使用的。我们再看看这个Foldable typeclass：scalaz/Foldable.scala
1 trait Foldable[F[_]] { self =>
2 ////
3 import collection.generic.CanBuildFrom
4 import collection.immutable.IndexedSeq
5
6 /** Map each element of the structure to a [[scalaz.Monoid]], and combine the results. */
7 def foldMap[A,B](fa: F[A])(f: A => B)(implicit F: Monoid[B]): B
8 /** As `foldMap` but returning `None` if the foldable is empty and `Some` otherwise */
9 def foldMap1Opt[A,B](fa: F[A])(f: A => B)(implicit F: Semigroup[B]): Option[B] = {
10 import std.option._
11 foldMap(fa)(x => some(f(x)))
12 }
13
14 /**Right-associative fold of a structure. */
15 def foldRight[A, B](fa: F[A], z: => B)(f: (A, => B) => B): B
16 ...
Foldable typeclass提供了许多注入方法支持折叠操作： scalaz/syntax/FoldableSyntax.scala
1 final class FoldableOps[F[_],A] private[syntax](val self: F[A])(implicit val F: Foldable[F]) extends Ops[F[A]] {
2 ////
3 import collection.generic.CanBuildFrom
4 import Leibniz.===
5 import Liskov.<~<
6
7 final def foldMap[B: Monoid](f: A => B = (a: A) => a): B = F.foldMap(self)(f)
8 final def foldMap1Opt[B: Semigroup](f: A => B = (a: A) => a): Option[B] = F.foldMap1Opt(self)(f)
9 final def foldRight[B](z: => B)(f: (A, => B) => B): B = F.foldRight(self, z)(f)
10 final def foldMapRight1Opt[B](z: A => B)(f: (A, => B) => B): Option[B] = F.foldMapRight1Opt(self)(z)(f)
11 final def foldRight1Opt(f: (A, => A) => A): Option[A] = F.foldRight1Opt(self)(f)
12 final def foldLeft[B](z: B)(f: (B, A) => B): B = F.foldLeft(self, z)(f)
13 final def foldMapLeft1Opt[B](z: A => B)(f: (B, A) => B): Option[B] = F.foldMapLeft1Opt(self)(z)(f)
14 final def foldLeft1Opt(f: (A, A) => A): Option[A] = F.foldLeft1Opt(self)(f)
15 final def foldRightM[G[_], B](z: => B)(f: (A, => B) => G[B])(implicit M: Monad[G]): G[B] = F.foldRightM(self, z)(f)
16 final def foldLeftM[G[_], B](z: B)(f: (B, A) => G[B])(implicit M: Monad[G]): G[B] = F.foldLeftM(self, z)(f)
17 final def foldMapM[G[_] : Monad, B : Monoid](f: A => G[B]): G[B] = F.foldMapM(self)(f)
18 final def fold(implicit A: Monoid[A]): A = F.fold(self)(A)
19 final def foldr[B](z: => B)(f: A => (=> B) => B): B = F.foldr(self, z)(f)
20 final def foldr1Opt(f: A => (=> A) => A): Option[A] = F.foldr1Opt(self)(f)
21 final def foldl[B](z: B)(f: B => A => B): B = F.foldl(self, z)(f)
22 final def foldl1Opt(f: A => A => A): Option[A] = F.foldl1Opt(self)(f)
23 final def foldrM[G[_], B](z: => B)(f: A => ( => B) => G[B])(implicit M: Monad[G]): G[B] = F.foldrM(self, z)(f)
24 final def foldlM[G[_], B](z: B)(f: B => A => G[B])(implicit M: Monad[G]): G[B] = F.foldlM(self, z)(f)
25 final def length: Int = F.length(self)
26 final def index(n: Int): Option[A] = F.index(self, n)
27 final def indexOr(default: => A, n: Int): A = F.indexOr(self, default, n)
28 final def sumr(implicit A: Monoid[A]): A = F.foldRight(self, A.zero)(A.append)
29 final def suml(implicit A: Monoid[A]): A = F.foldLeft(self, A.zero)(A.append(_, _))
30 final def toList: List[A] = F.toList(self)
31 final def toVector: Vector[A] = F.toVector(self)
32 final def toSet: Set[A] = F.toSet(self)
33 final def toStream: Stream[A] = F.toStream(self)
34 final def toIList: IList[A] = F.toIList(self)
35 final def toEphemeralStream: EphemeralStream[A] = F.toEphemeralStream(self)
36 final def to[G[_]](implicit c: CanBuildFrom[Nothing, A, G[A]]) = F.to[A, G](self)
37 final def all(p: A => Boolean): Boolean = F.all(self)(p)
38 final def ∀(p: A => Boolean): Boolean = F.all(self)(p)
39 final def allM[G[_]: Monad](p: A => G[Boolean]): G[Boolean] = F.allM(self)(p)
40 final def anyM[G[_]: Monad](p: A => G[Boolean]): G[Boolean] = F.anyM(self)(p)
41 final def any(p: A => Boolean): Boolean = F.any(self)(p)
42 final def ∃(p: A => Boolean): Boolean = F.any(self)(p)
43 final def count: Int = F.count(self)
44 final def maximum(implicit A: Order[A]): Option[A] = F.maximum(self)
45 final def maximumOf[B: Order](f: A => B): Option[B] = F.maximumOf(self)(f)
46 final def maximumBy[B: Order](f: A => B): Option[A] = F.maximumBy(self)(f)
47 final def minimum(implicit A: Order[A]): Option[A] = F.minimum(self)
48 final def minimumOf[B: Order](f: A => B): Option[B] = F.minimumOf(self)(f)
49 final def minimumBy[B: Order](f: A => B): Option[A] = F.minimumBy(self)(f)
50 final def longDigits(implicit d: A <:< Digit): Long = F.longDigits(self)
51 final def empty: Boolean = F.empty(self)
52 final def element(a: A)(implicit A: Equal[A]): Boolean = F.element(self, a)
53 final def splitWith(p: A => Boolean): List[NonEmptyList[A]] = F.splitWith(self)(p)
54 final def selectSplit(p: A => Boolean): List[NonEmptyList[A]] = F.selectSplit(self)(p)
55 final def collapse[X[_]](implicit A: ApplicativePlus[X]): X[A] = F.collapse(self)
56 final def concatenate(implicit A: Monoid[A]): A = F.fold(self)
57 final def intercalate(a: A)(implicit A: Monoid[A]): A = F.intercalate(self, a)
58 final def traverse_[M[_]:Applicative](f: A => M[Unit]): M[Unit] = F.traverse_(self)(f)
59 final def traverseU_[GB](f: A => GB)(implicit G: Unapply[Applicative, GB]): G.M[Unit] =
60 F.traverseU_[A, GB](self)(f)(G)
61 final def traverseS_[S, B](f: A => State[S, B]): State[S, Unit] = F.traverseS_(self)(f)
62 final def sequence_[G[_], B](implicit ev: A === G[B], G: Applicative[G]): G[Unit] = F.sequence_(ev.subst[F](self))(G)
63 final def sequenceS_[S, B](implicit ev: A === State[S,B]): State[S,Unit] = F.sequenceS_(ev.subst[F](self))
64 def sequenceF_[M[_],B](implicit ev: F[A] <~< F[Free[M,B]]): Free[M, Unit] = F.sequenceF_(ev(self))
65 final def msuml[G[_], B](implicit ev: A === G[B], G: PlusEmpty[G]): G[B] = F.foldLeft(ev.subst[F](self), G.empty[B])(G.plus[B](_, _))
66 ////
67 }
这简直就是一个完整的函数库嘛。scalaz为大多数标准库中的集合类型提供了Foldable实例，也就是说大多数scala集合类型都支持这么一堆折叠操作函数。我还看不到任何需要去自定义集合类型，标准库的集合类型加上Foldable typeclass应该足够用了。
在Foldable typeclass中比较重要的函数就是foldMap了：
1 trait Foldable[F[_]] { self =>
2 ////
3 import collection.generic.CanBuildFrom
4 import collection.immutable.IndexedSeq
5
6 /** Map each element of the structure to a [[scalaz.Monoid]], and combine the results. */
7 def foldMap[A,B](fa: F[A])(f: A => B)(implicit F: Monoid[B]): B
首先，foldMap需要Monoid[B]实例来实现。用List来举例：List trait 继承了Traverse：scalaz/std/List.scala
1 trait ListInstances extends ListInstances0 {
2 implicit val listInstance = new Traverse[List] with MonadPlus[List] with Zip[List] with Unzip[List] with Align[List] with IsEmpty[List] with Cobind[List] {
3 ...
在Traverse typeclass里定义了Foldable实例：scalaz/Traverse.scala
1 def foldLShape[A,B](fa: F[A], z: B)(f: (B,A) => B): (B, F[Unit]) =
2 runTraverseS(fa, z)(a => State.modify(f(_, a)))
3
4 override def foldLeft[A,B](fa: F[A], z: B)(f: (B,A) => B): B = foldLShape(fa, z)(f)._1
5
6 def foldMap[A,B](fa: F[A])(f: A => B)(implicit F: Monoid[B]): B = foldLShape(fa, F.zero)((b, a) => F.append(b, f(a)))._1
7
8 override def foldRight[A, B](fa: F[A], z: => B)(f: (A, => B) => B) =
9 foldMap(fa)((a: A) => (Endo.endo(f(a, _: B)))) apply z
10 ...
这个foldMap就是一个游览可折叠结构的函数。在游览过程中用Monoid append对结构中元素进行操作。值得注意的是这个f: A => B参数：这个函数是用来在append操作之前先对内部元素进行一次转变（transform）：
1 List(1,2,3) foldMap {x => x} //> res18: Int = 6
2 List(1,2,3) foldMap {x => (x + 3).toString} //> res19: String = 456 变成String操作
我们试着用一些实际的例子来示范Monoid的用法。上面提到Monoid在可折叠数据结构里的元素连续处理有着很好的应用，我们先试一个例子：确定一个可折叠数据结构F[A]中的元素A是否排序的：
def ordered(xs: List[Int]): Boolean //判断xs是否按序排列
由于我们必须游览List xs，所以用Monoid对元素Int进行判断操作是可行的方法。我们先设计一个对比数据结构：
Option[(min: Int, max: Int. ordered: Boolean)], 它记录了当前元素的状态，包括最小，最大，是否排序的：
1 /判断xs是否是排序的
2 def ordered(xs: List[Int]): Boolean = {
3 val monoid = new Monoid[Option[(Int,Int,Boolean)]] { //对类型Option[(Int,Int,Boolean)]定义一个Monoid实例
4 def zero = None
5 def append(a1: Option[(Int,Int,Boolean)], a2: => Option[(Int,Int,Boolean)]) = //对连续两个元素进行对比操作
6 (a1,a2) match {
7 case (x,None) => x
8 case (None,x) => x //保留不为None的状态
9 case (Some((min1,max1,ord1)),Some((min2,max2,ord2))) => //如果max1 <= min2状态即为true
10 Some((min1 min min2, max1 max max2, ord1 && ord2 && (max1 <= min2))) //更新min,max和ord
11 }
12 } //我们需要把元素转换成Option((Int,Int,Boolean))
13 (xs.foldMap(i => Option((i, i, true)))(monoid)).map(_._3) getOrElse(true)
14 } //> ordered: (xs: List[Int])Boolean
15
16 ordered(List(1,2,12,34)) //> res21: Boolean = true
17 ordered(List(1,2,34,23)) //> res22: Boolean = false
注意这个i => Option((i,i,true)) 转换（transform）。
由于Monoid是种极简单的类型，所以很容易对Monoid进行组合。Monoid组合产生的结果还是Monoid，并且用起来可以更方便：
1 def productMonoid[A,B](ma: Monoid[A], mb: Monoid[B]): Monoid[(A,B)] = new Monoid[(A,B)] {
2 def zero = (ma.zero, mb.zero)
3 def append(x: (A,B), y: => (A,B)): (A,B) = (ma.append(x._1, y._1), mb.append(x._2, y._2))
4 } //> productMonoid: [A, B](ma: scalaz.Monoid[A], mb: scalaz.Monoid[B])scalaz.Mon
5 //| oid[(A, B)]
6 val pm = productMonoid(Monoid[Int],Monoid[List[Int]])
7 //> pm : scalaz.Monoid[(Int, List[Int])] = Exercises.monoid$$anonfun$main$1$$a
8 //| non$3@72d1ad2e
以上的pm就是两个Monoid的组合，结果是一个tuple2Monoid。我们可以使用这个tuple2Monoid对可折叠数据结构中元素进行并行操作。比如我们可以在游览一个List[Int]时同时统计长度（list length）及乘积（product）:
1 val intMultMonoid = new Monoid[Int] {
2 def zero = 1
3 def append(a1: Int, a2: => Int): Int = a1 * a2
4 } //> intMultMonoid : scalaz.Monoid[Int] = Exercises.monoid$$anonfun$main$1$$ano
5 //| n$1@6c64cb25
6 def productMonoid[A,B](ma: Monoid[A], mb: Monoid[B]): Monoid[(A,B)] = new Monoid[(A,B)] {
7 def zero = (ma.zero, mb.zero)
8 def append(x: (A,B), y: => (A,B)): (A,B) = (ma.append(x._1, y._1), mb.append(x._2, y._2))
9 } //> productMonoid: [A, B](ma: scalaz.Monoid[A], mb: scalaz.Monoid[B])scalaz.Mon
10 //| oid[(A, B)]
11 val pm = productMonoid(Monoid[Int @@ Tags.Multiplication],Monoid[Int])
12 //> pm : scalaz.Monoid[(scalaz.@@[Int,scalaz.Tags.Multiplication], Int)] = Exe
13 //| rcises.monoid$$anonfun$main$1$$anon$3@72d1ad2e
14 List(1,2,3,4,6).foldMap(i => (i, 1))(productMonoid(intMultMonoid,Monoid[Int]))
15 //> res23: (Int, Int) = (144,5)
我们再来一个合并多层map的Monoid:
1 def mapMergeMonoid[K,V](V: Monoid[V]): Monoid[Map[K, V]] =
2 new Monoid[Map[K, V]] {
3 def zero = Map[K,V]()
4 def append(a: Map[K, V], b: => Map[K, V]) =
5 (a.keySet ++ b.keySet).foldLeft(zero) { (acc,k) =>
6 acc.updated(k, V.append(a.getOrElse(k, V.zero),
7 b.getOrElse(k, V.zero)))
8 }
9 } //> mapMergeMonoid: [K, V](V: scalaz.Monoid[V])scalaz.Monoid[Map[K,V]]
10
11 val M: Monoid[Map[String, Map[String, Int]]] = mapMergeMonoid(mapMergeMonoid(Monoid[Int]))
12 //> M : scalaz.Monoid[Map[String,Map[String,Int]]] = Exercises.monoid$$anonfun
13 //| $main$1$$anon$4@79e2c065
14 val m1 = Map("o1" -> Map("i1" -> 1, "i2" -> 2)) //> m1 : scala.collection.immutable.Map[String,scala.collection.immutable.Map[
15 //| String,Int]] = Map(o1 -> Map(i1 -> 1, i2 -> 2))
16 val m2 = Map("o1" -> Map("i2" -> 3)) //> m2 : scala.collection.immutable.Map[String,scala.collection.immutable.Map[
17 //| String,Int]] = Map(o1 -> Map(i2 -> 3))
18 val m3 = M.append(m1, m2) //> m3 : Map[String,Map[String,Int]] = Map(o1 -> Map(i1 -> 1, i2 -> 5))
我们可以用这个组合成的M的append操作进行map的深度合并。m1,m2合并后：Map(o1->Map("i1"->1,"i2" -> 5))。
我们还可以用这个Monoid来统计一段字串内字符发生的频率：
1 def frequencyMap[A](as: List[A]): Map[A, Int] =
2 as.foldMap((a: A) => Map(a -> 1))(mapMergeMonoid[A, Int](Monoid[Int]))
3 //> frequencyMap: [A](as: List[A])Map[A,Int]
4 frequencyMap("the brown quik fox is running quikly".toList)
5 //> res24: Map[Char,Int] = Map(e -> 1, s -> 1, x -> 1, n -> 4, y -> 1, t -> 1,
6 //| u -> 3, f -> 1, i -> 4, -> 6, q -> 2, b -> 1, g -> 1, l -> 1, h -> 1, r -
7 //| > 2, w -> 1, k -> 2, o -> 2)
我们现在可以体会到Monoid必须在可折叠数据结构（Foldable）内才能正真发挥作用。