algebra

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 BoundedSemilattice[A] extends Semilattice[A] with CommutativeMonoid[A]

   

3. trait CommutativeGroup[A] extends Group[A] with CommutativeMonoid[A]

   

   An abelian group is a group whose operation is commutative.

4. 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.

5. 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.

6. 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.

7. trait EqFunctions extends AnyRef

   

8. trait Group[A] extends Monoid[A]

   

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

9. trait GroupFunctions[G[T] <: Group[T]] extends MonoidFunctions[Group]

   

10. 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 = "".

11. trait MonoidFunctions[M[T] <: Monoid[T]] extends SemigroupFunctions[M]

    

12. 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.

13. trait OrderFunctions extends AnyRef

    

14. 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 Double 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)

15. trait PartialOrderFunctions extends AnyRef

    

16. sealed trait Priority[+P, +F] extends AnyRef

    

    Priority is a type class for prioritized implicit search.

    Priority is a type class for prioritized implicit search.

    This type class will attempt to provide an implicit instance of P (the preferred type). If that type is not available it will fallback to F (the fallback type). If neither type is available then a Priority[P, F] instance will not be available.

    This type can be useful for problems where multiple algorithms can be used, depending on the type classes available.

17. trait Semigroup[A] extends Serializable

    

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

18. trait SemigroupFunctions[S[T] <: Semigroup[T]] extends AnyRef

    

19. 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.