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)
   

   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.

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

   The associative law states that given values a, b, and c of type A (and implicit Associative.adapters imported):

   ((a combine b) combine c) === (a combine (b combine c))
   

   Note: In mathematics, the algebraic structure consisting of a set along with an associative binary operation is known as a semigroup.

3. trait Commutative[A] extends AnyRef

   

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

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

   The commutative law states that changing the order of the operands to a binary operation does not change the result, i.e. given values a, b

   (a combine b) === (b combine a)
   

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.

   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.

   The distributive law states that given values a, b, c

   a dcombine (b combine c) === (a dcombine b) combine (a dcombine c)
   

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

   

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

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

   The identity law states that given a value ca of type Context[A] and the identity function, (a: A) => a (and implicit Functor.adapters imported):

   ca.map((a: A) => a) === ca
   

   The composition law states that given a value ca of type Context[A] and two functions, f of type A => B and g of type B => C (and implicit Functor.adapters imported):

   ca.map(f).map(g) === ca.map(g compose f)
   

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.

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

   A Monad instance wraps an object that in some way behaves as a Monad.

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.

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

   The associative law states that given values a, b, and c of type A (and implicit Monoid.adapters imported):

   ((a combine b) combine c) === (a combine (b combine c))
   

   The left identity law states that given the identity value, z, and any other value, a, of type A (and implicit Monoid.adapters imported):

   (z combine a) === a
   

   An similarly, the right identity law states that given the same values and implicit:

   (a combine z) === a