Type Members

1.  trait Band[A] extends Semigroup[A]

   Bands are semigroups whose operation (i.e.

   Bands are semigroups whose operation (i.e. combine) is also idempotent.

2.  trait BoundedEnumerable[A] extends PartialPreviousUpperBounded[A] with PartialNextLowerBounded[A]
3.  trait BoundedSemilattice[A] extends Semilattice[A] with CommutativeMonoid[A]
4.  trait CommutativeGroup[A] extends Group[A] with CommutativeMonoid[A]

   An commutative group (also known as an abelian group) is a group whose combine operation is commutative.

5.  trait CommutativeMonoid[A] extends Monoid[A] with CommutativeSemigroup[A]

   CommutativeMonoid represents a commutative monoid.

   CommutativeMonoid represents a commutative monoid.

   A monoid is commutative if for all x and y, x |+| y === y |+| x.

6.  trait CommutativeSemigroup[A] extends Semigroup[A]

   CommutativeSemigroup represents a commutative semigroup.

   CommutativeSemigroup represents a commutative semigroup.

   A semigroup is commutative if for all x and y, x |+| y === y |+| x.

7.  sealed abstract class Comparison extends Product with Serializable

   ADT encoding the possible results of a comparison

8.  trait Eq[A] extends Serializable

   A type class used to determine equality between 2 instances of the same type.

   A type class used to determine equality between 2 instances of the same type. Any 2 instances x and y are equal if eqv(x, y) is true. Moreover, eqv should form an equivalence relation.

9.  abstract class EqFunctions[E[T] <: Eq[T]] extends AnyRef
10.  trait EqToEquivConversion extends AnyRef
11.  trait Group[A] extends Monoid[A]

    A group is a monoid where each element has an inverse.

12.  abstract class GroupFunctions[G[T] <: Group[T]] extends MonoidFunctions[G]
13.  trait Hash[A] extends Eq[A] with Serializable

    A type class used to represent a hashing scheme for objects of a given type.

    A type class used to represent a hashing scheme for objects of a given type. For any two instances x and y that are considered equivalent under the equivalence relation defined by this object, hash(x) should equal hash(y).

14.  abstract class HashFunctions[H[T] <: Hash[T]] extends EqFunctions[H]
15.  trait HashToHashingConversion extends AnyRef
16.  trait LowerBounded[A] extends AnyRef

    A type class used to name the lower limit of a type.

17.  trait LowerBoundedEnumerable[A] extends PartialNextLowerBounded[A] with Next[A]
18.  trait LowerBoundedFunctions[L[T] <: LowerBounded[T]] extends AnyRef
19.  trait Monoid[A] extends Semigroup[A]

    A monoid is a semigroup with an identity.

    A monoid is a semigroup with an identity. A monoid is a specialization of a semigroup, so its operation must be associative. Additionally, combine(x, empty) == combine(empty, x) == x. For example, if we have Monoid[String], with combine as string concatenation, then empty = "".

20.  abstract class MonoidFunctions[M[T] <: Monoid[T]] extends SemigroupFunctions[M]

    Annotations

    @suppressUnusedImportWarningForScalaVersionSpecific()

21.  trait Next[A] extends PartialNext[A]

    A typeclass with an operation which returns a member which is always greater than the one supplied.

22.  trait Order[A] extends PartialOrder[A]

    The Order type class is used to define a total ordering on some type A.

    The Order type class is used to define a total ordering on some type A. An order is defined by a relation <=, which obeys the following laws:

    - either x <= y or y <= x (totality) - if x <= y and y <= x, then x == y (antisymmetry) - if x <= y and y <= z, then x <= z (transitivity)

    The truth table for compare is defined as follows:

    x <= y x >= y Int true true = 0 (corresponds to x == y) true false < 0 (corresponds to x < y) false true > 0 (corresponds to x > y)

    By the totality law, x <= y and y <= x cannot be both false.

23.  abstract class OrderFunctions[O[T] <: Order[T]] extends PartialOrderFunctions[O]
24.  trait OrderToOrderingConversion extends AnyRef
25.  trait PartialNext[A] extends AnyRef

    A typeclass with an operation which returns a member which is greater or None than the one supplied.

26.  trait PartialNextLowerBounded[A] extends PartialPrevious[A] with PartialNext[A] with LowerBounded[A]
27.  trait PartialOrder[A] extends Eq[A]

    The PartialOrder type class is used to define a partial ordering on some type A.

    The PartialOrder type class is used to define a partial ordering on some type A.

    A partial order is defined by a relation <=, which obeys the following laws:

    • x <= x (reflexivity)
    • if x <= y and y <= x, then x = y (anti-symmetry)
    • if x <= y and y <= z, then x <= z (transitivity)

    To compute both <= and >= at the same time, we use a Double number to encode the result of the comparisons x <= y and x >= y. The truth table is defined as follows:

    x <= y	x >= y	result	note
    true	true	0.0	(corresponds to x = y)
    false	false	NaN	(x and y cannot be compared)
    true	false	-1.0	(corresponds to x < y)
    false	true	1.0	(corresponds to x > y)

28.  abstract class PartialOrderFunctions[P[T] <: PartialOrder[T]] extends EqFunctions[P]
29.  trait PartialOrderToPartialOrderingConversion extends AnyRef
30.  trait PartialPrevious[A] extends AnyRef

    A typeclass with an operation which returns a member which is smaller or None than the one supplied.

31.  trait PartialPreviousUpperBounded[A] extends PartialPrevious[A] with PartialNext[A] with UpperBounded[A]
32.  trait Previous[A] extends PartialPrevious[A]

    A typeclass with an operation which returns a member which is always smaller than the one supplied.

33.  trait Semigroup[A] extends Serializable

    A semigroup is any set A with an associative operation (combine).

34.  abstract class SemigroupFunctions[S[T] <: Semigroup[T]] extends AnyRef
35.  trait Semilattice[A] extends Band[A] with CommutativeSemigroup[A]

    Semilattices are commutative semigroups whose operation (i.e.

    Semilattices are commutative semigroups whose operation (i.e. combine) is also idempotent.

36.  abstract class SemilatticeFunctions[S[T] <: Semilattice[T]] extends SemigroupFunctions[S]
37.  trait UnboundedEnumerable[A] extends Next[A] with Previous[A]

    A typeclass which has both previous and next operations such that next . previous == identity.

38.  trait UpperBounded[A] extends AnyRef

    A type class used to name the upper limit of a type.

39.  trait UpperBoundedEnumerable[A] extends PartialPreviousUpperBounded[A] with Previous[A]
40.  trait UpperBoundedFunctions[U[T] <: UpperBounded[T]] extends AnyRef