algebra

Type Members

1. trait Applicative[Context[_]] extends Functor[Context]

   Typeclass trait representing an algebraic structure that is a Functor[Context] (i.e., it declares a map method that obeys the functor laws) augmented by insert and applying methods that obey laws of homomorphism and interchange. The homomorphism law states that given a value, a, of type A and a function, f, of type A => B (and implicit Applicative.adapters imported):

   val ca = applicative.insert(a)
   val cf = applicative.insert(f)
   ca.applying(cf) === applicative.insert(f(a))
   

   The interchange law states that given a value, a, of type A and a function, cf, of type Context[A => B] (and implicit Applicative.adapters imported):

   val ca = applicative.insert(a)
   val cg = applicative.insert((f: A => B) => f(a))
   ca.applying(cf) === cf.applying(cg)
   

2. trait Associative[A] extends AnyRef

   Typeclass trait representing a binary operation that obeys the associative law.

3. trait Commutative[A] extends AnyRef

   Typeclass trait representing a binary operation that obeys the commutative law.

4. trait Distributive[A] extends AnyRef

   Typeclass trait representing two binary operations that obeys the distributive law: one, dcombine that does the distributing, and the other combine binary combine that dcombine is applied to.

5. trait Functor[Context[_]] extends AnyRef

   Typeclass trait representing an algebraic structure defined by a map method that obeys laws of identity and composition.

6. trait Monad[Context[_]] extends Applicative[Context]

   Typeclass trait for algebraic structure containing insertion and flat-mapping methods that obey laws of identity and associativity.

7. trait Monoid[A] extends Associative[A]

   Typeclass trait representing a binary operation that obeys the associative law and an identity element that obeys the left and right identity laws.