Understanding Monad 1 - Group and Monoid

Group

If we define a binary operator $op$ on set $G$, which satisfied such properties

  • Closed, for any $a$ and $b$, has property $op(a,b) \in G$

  • Associativity, for any $a$ and $b$, has property $op(op(a, b), c) = op(a, op(b, c))$

  • Exist an identity element $e$ which for any $a$, has $op(e, a) = op(a, e) = a$

  • Exist inverse element, for any $a \in G$, has a inverse element $a^{-1}$ which $ op(a^{-1}, a) = op(a, a^{-1}) = e $

then we say $(G, op)$ is a group.

Semigroup

If a binary operation and set combination $(G, op)$ just satisfy closure and associativity properties, then we call it is a semigroup.

Monoid

If a group $(G, op)$ satisfies closure, associativity, and also exists an identity, then we call it is a monoid.

Example

For examples, we can use Scale code to demonstrate these concepts.

trait SemiGroup[T] {
    def plus(a : T, b : T) : T
}

// Int is a semigroup
object IntSemiGroup extends SemiGroup[Int] {
    def plus(a : Int, b : Int) = a + b
}

trait Monoid[T] extends SemiGroup[T] {
    def identity : T
}

// Int is also a Monoid
object IntMonoid extends Monoid[Int] {
    def plus(a : Int, b : Int) = a + b
    def identity = 0
}