Monoid

Monoid instances must satisfy scalaz.Semigroup.SemigroupLaw and 2 additional laws:

• '''left identity''': forall a. append(zero, a) == a
• '''right identity''' : forall a. append(a, zero) == a

A semigroup in type F must satisfy two laws:

• '''closure''': ∀ a, b in F, append(a, b) is also in F. This is enforced by the type system.
• '''associativity''': ∀ a, b, c in F, the equation append(append(a, b), c) = append(a, append(b , c)) holds.

The identity element for append.

A monoidal applicative functor, that implements point and ap with the operations zero and append respectively. Note that the type parameter α in Applicative[λ[α => F]] is discarded; it is a phantom type. As such, the functor cannot support scalaz.Bind.

Every Monoid gives rise to a scalaz.Category, for which the type parameters are phantoms.

Whether a == zero.

For n = 0, zero For n = 1, append(zero, value) For n = 2, append(append(zero, value), value)

The binary operation to combine f1 and f2.

Implementations should not evaluate the by-name parameter f2 if result can be determined by f1.

An scalaz.Apply, that implements ap with append. Note that the type parameter α in Apply[λ[α => F]] is discarded; it is a phantom type. As such, the functor cannot support scalaz.Bind.

Every Semigroup gives rise to a scalaz.Compose, for which the type parameters are phantoms.

For n = 0, value For n = 1, append(value, value) For n = 2, append(append(value, value), value)

The default definition uses peasant multiplication, exploiting associativity to only require O(log n) uses of append