IsomorphismMonoid

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.

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.

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

Every Semigroup gives rise to a scalaz.Compose, 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)

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

Unfold seed to the left and sum using append. Semigroups with right absorbing elements may override this method to not unfold more than is necessary to determine the result.

Unfold seed to the right and sum using append. Semigroups with left absorbing elements may override this method to not unfold more than is necessary to determine the result.