Scala Design Patterns - Second Edition

Monoids and composition

So far, we have seen some examples where monoids are used to improve efficiency and write generic functions. They, however, are even more powerful. The reason is that they follow another useful rule:

Monoids support composition; if A and B are monoids, then their product (A, B) is also a monoid.

What does this mean exactly and how can we take advantage of this? Let's look at the following function:

def compose[T, Y](a: Monoid[T], b: Monoid[Y]): Monoid[(T, Y)] =  new Monoid[(T, Y)] {    val zero: (T, Y) = (a.zero, b.zero)    override def op(l: (T, Y), r: (T, Y)): (T, Y) =      (a.op(l._1, r._1), b.op(l._2, r._2))  }

In the preceding code, we showed a function that applies the composition exactly as we mentioned in our definition. ...