Semigroup 

If you have the book Learn You a Haskell for Great Good you get to start a new chapter: “Monoids.” For the website, it’s still Functors, Applicative Functors and Monoids.

First, it seems like Cats is missing newtype/tagged type facility. We’ll implement our own later.

Haskell’s Monoid is split into Semigroup and Monoid in Cats. They are also type aliases of algebra.Semigroup and algebra.Monoid. As with Apply and Applicative, Semigroup is a weaker version of Monoid. If you can solve the same problem, weaker is cooler because you’re making fewer assumptions.

LYAHFGG:

It doesn’t matter if we do (3 * 4) * 5 or 3 * (4 * 5). Either way, the result is 60. The same goes for ++. …

We call this property associativity. * is associative, and so is ++, but -, for example, is not.

Let’s check this:

import cats._, cats.syntax.all._

assert { (3 * 2) * (8 * 5) === 3 * (2 * (8 * 5)) }

assert { List("la") ++ (List("di") ++ List("da")) === (List("la") ++ List("di")) ++ List("da") }

No error means, they are equal.

The Semigroup typeclass 

Here’s the typeclass contract for algebra.Semigroup.

/**
 * A semigroup is any set `A` with an associative operation (`combine`).
 */
trait Semigroup[@sp(Int, Long, Float, Double) A] extends Any with Serializable {

  /**
   * Associative operation taking which combines two values.
   */
  def combine(x: A, y: A): A

  ....
}

This enables combine operator and its symbolic alias |+|. Let’s try using this.

List(1, 2, 3) |+| List(4, 5, 6)
// res2: List[Int] = List(1, 2, 3, 4, 5, 6)

"one" |+| "two"
// res3: String = "onetwo"

The Semigroup Laws 

Associativity is the only law for Semigroup.

• associativity (x |+| y) |+| z = x |+| (y |+| z)

Here’s how we can check the Semigroup laws from the REPL. Review Checking laws with discipline for the details:

scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._

scala> import cats.kernel.laws.GroupLaws
import cats.kernel.laws.GroupLaws

scala> val rs1 = GroupLaws[Int].semigroup(Semigroup[Int])
rs1: cats.kernel.laws.GroupLaws[Int]#GroupProperties = cats.kernel.laws.GroupLaws$GroupProperties@5a077d1d

scala> rs1.all.check
+ semigroup.associativity: OK, passed 100 tests.
+ semigroup.combineN(a, 1) == a: OK, passed 100 tests.
+ semigroup.combineN(a, 2) == a |+| a: OK, passed 100 tests.
+ semigroup.serializable: OK, proved property.

Lists are Semigroups 

List(1, 2, 3) |+| List(4, 5, 6)
// res4: List[Int] = List(1, 2, 3, 4, 5, 6)

Product and Sum 

For Int a semigroup can be formed under both + and *. Instead of tagged types, cats provides only the instance additive.

Trying to use operator syntax here is tricky.

def doSomething[A: Semigroup](a1: A, a2: A): A =
  a1 |+| a2

doSomething(3, 5)(Semigroup[Int])
// res5: Int = 8

I might as well stick to function syntax:

Semigroup[Int].combine(3, 5)
// res6: Int = 8