algebra

Type Members

1. trait AbGroup[A] extends Group[A] with CMonoid[A]

   An abelian group is a group whose operation is commutative.

2. trait Action[P, G] extends LeftAction[P, G] with RightAction[P, G]

   A semigroup/monoid/group action of G on P is the combination of compatible left and right actions, providing:

3. trait AdditiveAbGroup[A] extends AdditiveGroup[A] with AdditiveCMonoid[A]

4. trait AdditiveAction[P, G] extends Any

5. trait AdditiveCMonoid[A] extends AdditiveMonoid[A] with AdditiveCSemigroup[A]

6. trait AdditiveCSemigroup[A] extends AdditiveSemigroup[A]

7. trait AdditiveGroup[A] extends AdditiveMonoid[A]

8. trait AdditiveMonoid[A] extends AdditiveSemigroup[A]

9. trait AdditiveSemigroup[A] extends Any

10. trait AdditiveTorsor[V, R] extends AdditiveAction[V, R]

11. trait Bool[A] extends Heyting[A]

    A boolean algebra is a structure that defines a few basic operations, namely as conjunction (&), disjunction (|), and negation (~).

12. trait CMonoid[A] extends Monoid[A] with CSemigroup[A]

    CMonoid represents a commutative monoid.

13. trait CRig[A] extends Rig[A] with MultiplicativeCMonoid[A]

    CRig is a Rig that is commutative under multiplication.

14. trait CRing[A] extends Ring[A] with MultiplicativeCMonoid[A]

    CRing is a Ring that is commutative under multiplication.

15. trait CSemigroup[A] extends Semigroup[A]

    CSemigroup represents a commutative semigroup.

16. trait CoordinateSpace[V, F] extends InnerProductSpace[V, F]

17. class DualBool[A] extends Bool[A]

18. trait Eq[A] extends Any

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

19. trait EuclideanRing[A] extends CRing[A]

20. trait Field[A] extends EuclideanRing[A] with MultiplicativeAbGroup[A]

21. trait FieldAlgebra[V, F] extends RingAlgebra[V, F] with VectorSpace[V, F]

    A FieldAlgebra is a vector space that is also a Ring.

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

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

23. trait InnerProductSpace[V, F] extends VectorSpace[V, F]

24. trait IsAlgebraic[A] extends IsReal[A]

25. trait IsIntegral[A] extends IsRational[A]

26. trait IsRational[A] extends IsAlgebraic[A]

27. trait IsReal[A] extends Order[A] with Signed[A]

    A simple type class for numeric types that are a subset of the reals.

28. trait LeftAction[P, G] extends Any

    A (left) semigroup/monoid/group action of G on P is simply the implementation of a method actl(g, p), or g |+|> p, such that:

29. trait MetricSpace[V, R] extends Any

    This type class models a metric space V.

30. trait Module[V, R] extends AdditiveAbGroup[V]

    A module generalizes a vector space by requiring its scalar need only form a ring, rather than a field.

31. trait Monoid[A] extends Semigroup[A]

    A monoid is a semigroup with an identity.

32. trait MultiplicativeAbGroup[A] extends MultiplicativeGroup[A] with MultiplicativeCMonoid[A]

33. trait MultiplicativeAction[P, G] extends Any

34. trait MultiplicativeCMonoid[A] extends MultiplicativeMonoid[A] with MultiplicativeCSemigroup[A]

35. trait MultiplicativeCSemigroup[A] extends MultiplicativeSemigroup[A]

36. trait MultiplicativeGroup[A] extends MultiplicativeMonoid[A]

37. trait MultiplicativeMonoid[A] extends MultiplicativeSemigroup[A]

38. trait MultiplicativeSemigroup[A] extends Any

39. trait MultiplicativeTorsor[V, R] extends MultiplicativeAction[V, R]

40. trait NRoot[A] extends Any

    This is a type class for types with n-roots.

41. trait NormedVectorSpace[V, F] extends VectorSpace[V, F] with MetricSpace[V, F]

    A normed vector space is a vector space equipped with a function norm: V => F.

42. trait Order[A] extends PartialOrder[A]

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

43. trait PartialOrder[A] extends Eq[A]

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

44. trait Rig[A] extends Semiring[A] with AdditiveMonoid[A] with MultiplicativeMonoid[A]

    Rig is a ring whose additive structure doesn't have an inverse (ie.

45. trait RightAction[P, G] extends Any

    A (right) semigroup/monoid/group action of G on P is simply the implementation of a method actr(p, g), or p <|+| g, such that:

46. trait Ring[A] extends Rig[A] with Rng[A]

    Ring represents a set (A) that is a group over addition (+) and a monoid over multiplication (*).

47. trait RingAlgebra[V, R] extends Module[V, R] with Rng[V]

    A RingAlgebra is a module that is also a Rng.

48. trait Rng[A] extends Semiring[A] with AdditiveAbGroup[A]

    Rng is a ring whose multiplicative structure doesn't have an identity (i.

49. trait Semigroup[A] extends Any

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

50. trait Semiring[A] extends AdditiveMonoid[A] with MultiplicativeSemigroup[A]

    Semiring is a ring without identities or an inverse.

51. sealed abstract class Sign extends AnyRef

    A simple ADT representing the Sign of an object.

52. trait Signed[A] extends Any

    A trait for things that have some notion of sign and the ability to ensure something has a positive sign.

53. trait Torsor[V, R] extends Action[V, R]

    A Torsor[V, R] requires an AbGroup[R] and provides Action[V, R], plus a diff operator, <-> in additive notation, such that:

54. trait Trig[A] extends Any

55. trait VectorSpace[V, F] extends Module[V, F]

    A vector space is a group V that can be multiplied by scalars in F that lie in a field.

56. trait ZAlgebra[V] extends RingAlgebra[V, Int] with Ring[V]

    Given any Ring[A] we can construct a RingAlgebra[A, Int].

57. final case class ZModule[V](vector: Group[V]) extends Module[V, Int] with Product with Serializable