Type Members

1.  abstract class AbstractEq [-X] extends Eq[X]
2.  abstract class AbstractHashing [-X] extends Hashing[X]
3.  abstract class AbstractOrder [-X] extends Order[X]
4.  abstract class AbstractPartialOrder [-X] extends PartialOrder[X]
5.  trait Action [X, -S] extends AnyRef

   Represents an action.

   Represents an action.

   Since

   0.2.10

6.  trait AdditiveAction [X, S] extends AnyRef

   Represents an action in which its action operator is translation (:+).

   Represents an action in which its action operator is translation (:+).

   Since

   0.2.10

7.  trait AdditiveCGroup [X] extends AdditiveGroup[X] with AdditiveCMonoid[X]

   Represents an additive Abelian (commutative) group.

   Represents an additive Abelian (commutative) group.

   An instance of this typeclass should satisfy the following axioms:

   • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
   • Additive identity: ∀a∈X, a + 0 == 0 + a == a.
   • Additive invertibility: \(\forall a\in X, \exists -a \in X, a + (-a) == (-a) + a == 0\).
   • Additive commutativity: ∀a, b∈X, a + b == b + a.

   Since

   0.2.6

8.  trait AdditiveCMonoid [X] extends AdditiveMonoid[X] with AdditiveCSemigroup[X]

   Represents an additive commutative monoid.

   Represents an additive commutative monoid.

   An instance of this typeclass should satisfy the following axioms:

   • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
   • Additive identity: ∀a∈X, a + 0 == 0 + a == a.
   • Additive commutativity: ∀a, b∈X, a + b == b + a.

   Since

   0.2.6

9.  trait AdditiveCSemigroup [X] extends AdditiveSemigroup[X]

   Represents an additive commutative semigroup.

   Represents an additive commutative semigroup.

   An instance of this typeclass should satisfy the following axioms:

   • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
   • Additive commutativity: ∀a, b∈X, a + b == b + a.
10.  trait AdditiveGroup [G] extends AdditiveMonoid[G]

    Represents an additive group.

    Represents an additive group. An additive group is an additive monoid (with operation + and identity element 0) that is also invertible.

    An instance of this typeclass should satisfy the following axioms:

    • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
    • Additive identity: ∀a∈X, a + 0 == 0 + a == a.
    • Additive invertibility: \(\forall a\in X, \exists -a \in X, a + (-a) == (-a) + a == 0\).
11.  trait AdditiveGroupAction [X, S] extends AdditiveMonoidAction[X, S]

    Represents an additive action where the actors form a group.

12.  trait AdditiveMonoid [M] extends AdditiveSemigroup[M] with HasZero[M]

    Represents an additive monoid.

    Represents an additive monoid. An additive monoid is an additive semigroup (with operation +) with an additive identity element 0.

    An instance of this typeclass should satisfy the following axioms:

    • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
    • Additive identity: ∀a∈X, a + 0 == 0 + a == a.

    Since

    0.1.0

13.  trait AdditiveMonoidAction [X, S] extends AdditiveSemigroupAction[X, S]

    Represents an additive action where the actors form a monoid.

14.  trait AdditiveSemigroup [S] extends AnyRef

    Represents an additive semigroup.

    Represents an additive semigroup. An additive semigroup is a semigroup with the binary operation add (+).

    An instance of this typeclass should satisfy the following axiom:

    • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
15.  trait AdditiveSemigroupAction [X, S] extends AdditiveAction[X, S]

    Represents an additive action where the actors form a semigroup.

16.  trait AdditiveTorsor [X, G] extends AdditiveGroupAction[X, G]
17.  trait AffineSpace [P, V, F] extends AdditiveTorsor[P, V]

    Represents an affine space.

    Represents an affine space.

    P

    Type of points

    V

    Type of vectors

    F

    Type of field on which the vector space is based

    Since

    0.2.10

18.  trait AlgebraOverField [X, S] extends VectorSpace[X, S] with AlgebraOverRing[X, S]

    Represents an algebra over a field, i.e., a vector space equipped with a bilinear product over the type of vectors.

    Represents an algebra over a field, i.e., a vector space equipped with a bilinear product over the type of vectors.

    Since

    0.1.0

19.  trait AlgebraOverRing [X, S] extends Module[X, S] with CRing[X]

    Represents an algebra over a ring, i.e., a module equipped with a bilinear product.

    Represents an algebra over a ring, i.e., a module equipped with a bilinear product.

    Since

    0.1.0

20.  trait BooleanAlgebra [X] extends BoundedLattice[X]

    Represents an abstract Boolean algebra, which is a complemented distributive lattice.

    Represents an abstract Boolean algebra, which is a complemented distributive lattice.

    Since

    0.1.0

21.  trait Bounded [+X] extends HasTop[X] with HasBottom[X]

    Witnesses the existence of both a maximum element and a minimum element of a specific type.

    Witnesses the existence of both a maximum element and a minimum element of a specific type.

    An instance of this typeclass should satisfy the following axioms:

    • Top element: ∀a∈X, a <= top.
    • Bottom element: ∀a∈X, a >= bot.

    Since

    0.2.7

22.  trait BoundedLattice [X] extends Lattice[X] with Bounded[X] with BoundedLowerSemilattice[X] with BoundedUpperSemilattice[X]

    Represents a bounded lattice, i.e., a lattice containing both the top and the bottom elements.

    Represents a bounded lattice, i.e., a lattice containing both the top and the bottom elements.

    An instance of this typeclass should satisfy the following axioms:

    • Supremum associativity: ∀a, b, c∈X, sup(a, sup(b, c)) == sup(sup(a, b), c).
    • Infimum associativity: ∀a, b, c∈X, inf(a, inf(b, c)) == inf(inf(a, b), c).
    • Supremum commutativity: ∀a, b∈X, sup(a, b) == sup(b, a).
    • Infimum commutativity: ∀a, b∈X, inf(a, b) == inf(b, a).
    • Absorption of supremum w.r.t. infimum: ∀a, b∈X, sup(a, inf(a, b)) == a.
    • Absorption of infimum w.r.t. supremum: ∀a, b∈X, inf(a, sup(a, b)) == a.
    • Top element: ∀a∈X, a <= top.
    • Bottom element: ∀a∈X, a >= bot.

    Since

    0.2.0

23.  trait BoundedLowerSemilattice [X] extends LowerSemilattice[X] with HasBottom[X]

    Represents a lower semilattice that has a specific bottom element.

    Represents a lower semilattice that has a specific bottom element.

    An instance of this typeclass should satisfy the following axioms:

    • Infimum associativity: ∀a, b, c∈X, inf(a, inf(b, c)) == inf(inf(a, b), c).
    • Infimum commutativity: ∀a, b∈X, inf(a, b) == inf(b, a).
    • Infimum idempotency: ∀a∈X, inf(a, a) == a.
    • Bottom element: ∀a∈X, a >= bot.

    Since

    0.2.0

24.  trait BoundedUpperSemilattice [X] extends UpperSemilattice[X] with HasTop[X]

    Represents an upper semilattice that has a specific top element.

    Represents an upper semilattice that has a specific top element.

    An instance of this typeclass should satisfy the following axioms:

    • Supremum associativity: ∀a, b, c∈X, sup(a, sup(b, c)) == sup(sup(a, b), c).
    • Supremum commutativity: ∀a, b∈X, sup(a, b) == sup(b, a).
    • Supremum idempotency: ∀a∈X, sup(a, a) == a.
    • Top element: ∀a∈X, a <= top.
25.  trait BoundedUpperSemilatticeWithEq [X] extends BoundedUpperSemilattice[X] with EqUpperSemilattice[X]
26.  trait CGroup [X] extends CMonoid[X] with Group[X]

    Represents an Abelian (commutative) group.

    Represents an Abelian (commutative) group.

    An instance of this typeclass should satisfy the following axioms:

    • Associativity: \(\forall a, b, c \in S, (ab)c = a(bc)\).
    • Identity: \( \forall a \in X, ia = ai = a\), where \(i\) is the identity element.
    • Invertibility: ∀a∈X, ∃b∈X, a op b == b op a == id.
    • Commutativity: \(\forall a, b \in X, ab = ba\).

    Since

    0.2.6

27.  trait CMonoid [X] extends Monoid[X] with CSemigroup[X]

    Represents a commutative monoid.

    Represents a commutative monoid.

    An instance of this typeclass should satisfy the following axioms:

    • Associativity: \(\forall a, b, c \in S, (ab)c = a(bc)\).
    • Identity: \( \forall a \in X, ia = ai = a\), where \(i\) is the identity element.
    • Commutativity: \(\forall a, b \in X, ab = ba\).

    Since

    0.2.6

28.  trait CRing [X] extends Ring[X] with CSemiring[X]

    Represents a commutative ring (i.e.

    Represents a commutative ring (i.e. the multiplication operation is also commutative).

    An instance of this typeclass should satisfy the following axioms:

    • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
    • Additive identity: ∀a∈X, a + 0 == 0 + a == a.
    • Additive invertibility: \(\forall a\in X, \exists -a \in X, a + (-a) == (-a) + a == 0\).
    • Additive commutativity: ∀a, b∈X, a + b == b + a.
    • Multiplicative associativity: ∀a, b, c∈X, (a * b) * c == a * (b * c).
    • Multiplicative identity: ∀a∈X, a * 1 == 1 * a == a.
    • Multiplicative commutativity: ∀a, b∈X, a * b == b * a.
    • Distributivity of multiplication w.r.t. addition: ∀a, b, c∈X, (a + b) * c == a * c + b * c and a * (b + c) == a * b + a * c.

    Since

    0.2.6

29.  trait CSemigroup [X] extends Semigroup[X]

    Represents a commutative semigroup.

    Represents a commutative semigroup.

    An instance of this typeclass should satisfy the following axioms:

    • Associativity: \(\forall a, b, c \in S, (ab)c = a(bc)\).
    • Commutativity: \(\forall a, b \in X, ab = ba\).
30.  trait CSemiring [X] extends Semiring[X] with MultiplicativeCMonoid[X]

    Represents a commutative semiring, i.e., a semiring where the multiplication operation is also commutative.

    Represents a commutative semiring, i.e., a semiring where the multiplication operation is also commutative.

    An instance of this typeclass should satisfy the following axioms:

    • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
    • Additive identity: ∀a∈X, a + 0 == 0 + a == a.
    • Additive commutativity: ∀a, b∈X, a + b == b + a.
    • Multiplicative associativity: ∀a, b, c∈X, (a * b) * c == a * (b * c).
    • Multiplicative identity: ∀a∈X, a * 1 == 1 * a == a.
    • Multiplicative commutativity: ∀a, b∈X, a * b == b * a.
    • Distributivity of multiplication w.r.t. addition: ∀a, b, c∈X, (a + b) * c == a * c + b * c and a * (b + c) == a * b + a * c.
    • Zero as an annihilator: ∀a∈X, 0 * a == a * 0 == 0.

    Since

    0.2.6

31.  trait ConcatenativeMonoid [X] extends ConcatenativeSemigroup[X] with HasEmpty[X]

    Represents a concatenative monoid (i.e.

    Represents a concatenative monoid (i.e. monoids operating on sequences, etc. that bears the operation ++)

    Since

    0.2.1

32.  trait ConcatenativeSemigroup [X] extends AnyRef

    Represents a concatenative semigroup (i.e.

    Represents a concatenative semigroup (i.e. semigroups operating on sequences, etc. that bears the operation ++)

    Since

    0.2.1

33.  trait Eq [-X] extends AnyRef

    Typeclass for type-strict equivalence relations.

    Typeclass for type-strict equivalence relations. This is equivalent to the Haskell typeclass Eq.

    Eq is contravariant with respect to its type parameter.

    An instance of this typeclass should satisfy the following axioms:

    • Reflexivity: ∀a∈X, a === a.
    • Symmetry: ∀a, b∈X, a === b implies b === a.
    • Transitivity: ∀a, b, c∈X, a === b and b === c implies a === c.
34.  trait EqBoundedLattice [X] extends EqBoundedLowerSemilattice[X] with BoundedUpperSemilatticeWithEq[X] with BoundedLattice[X]
35.  trait EqBoundedLowerSemilattice [X] extends BoundedLowerSemilattice[X] with EqLowerSemilattice[X]
36.  trait EqLattice [X] extends Lattice[X] with EqLowerSemilattice[X] with EqUpperSemilattice[X]
37.  trait EqLowerSemilattice [X] extends LowerSemilattice[X] with PartialOrder[X]
38.  trait EqUpperSemilattice [X] extends UpperSemilattice[X] with PartialOrder[X]
39.  trait EuclideanDomain [X] extends CRing[X] with OrderedRing[X]

    Represents a Euclidean domain.

    Represents a Euclidean domain.

    Since

    0.1.0

40.  trait Field [X] extends CRing[X] with MultiplicativeCGroup[X]

    Represents a field.

    Represents a field.

    Since

    0.1.0

41.  trait Group [G] extends Monoid[G]

    Represents a group.

    Represents a group. A group is a monoid where an inverse element exists for every element.

    An instance of this typeclass should satisfy the following axioms:

    • Associativity: \(\forall a, b, c \in S, (ab)c = a(bc)\).
    • Identity: \( \forall a \in X, ia = ai = a\), where \(i\) is the identity element.
    • Invertibility: ∀a∈X, ∃b∈X, a op b == b op a == id.
42.  trait GroupAction [X, S] extends MonoidAction[X, S]
43.  trait HasBottom [+X] extends AnyRef

    Represents the existence of a minimum bottom element.

    Represents the existence of a minimum bottom element.

    An instance of this typeclass should satisfy the following axiom:

    • Bottom element: ∀a∈X, a >= bot.
44.  trait HasEmpty [+X] extends AnyRef

    Represents the existence of an empty element.

45.  trait HasIdentity [+X] extends AnyRef

    Represents the existence of an identity element.

    Represents the existence of an identity element. An instance of this typeclass should satisfy the following axiom:

    • Identity: \( \forall a \in X, ia = ai = a\), where \(i\) is the identity element.
46.  trait HasOne [+X] extends AnyRef

    Represents the existence of an 1 element.

    Represents the existence of an 1 element.

    An instance of this typeclass should satisfy the following axiom:

    • Multiplicative identity: ∀a∈X, a * 1 == 1 * a == a.
47.  trait HasTop [+X] extends AnyRef

    Represents the existence of a maximum top element.

    Represents the existence of a maximum top element.

    An instance of this typeclass should satisfy the following axiom:

    • Top element: ∀a∈X, a <= top.
48.  trait HasZero [+X] extends AnyRef

    Represents the existence of an 0 element.

    Represents the existence of an 0 element.

    An instance of this typeclass should satisfy the following axiom:

    • Additive identity: ∀a∈X, a + 0 == 0 + a == a.
49.  trait Hashing [-X] extends Eq[X]

    Represents an equivalence relation endowed with a hashing function.

    Represents an equivalence relation endowed with a hashing function. This is the typeclass that a HashMap requires for its keys.

    Since

    0.2.0

50.  type Id[X] = X
51.  trait ImplicitStructures extends AnyRef
52.  trait InnerProductSpace [V, F] extends NormedVectorSpace[V, F]

    Typeclass for inner product spaces.

    Typeclass for inner product spaces. An inner product space is a vector space endowed with an inner product operation.

    Since

    0.1.0

53.  trait IsReal [X] extends OrderedField[X] with PowerOps[X] with TrigExpOps[X] with MetricSpace[X, X]

    Witnesses that a type represents the set of real numbers.

    Witnesses that a type represents the set of real numbers.

    Since

    0.2.10

54.  trait Lattice [X] extends UpperSemilattice[X] with LowerSemilattice[X]

    Represents a lattice, a partially-ordered set where each two elements have a unique infimum and a unique supremum.

    Represents a lattice, a partially-ordered set where each two elements have a unique infimum and a unique supremum.

    An instance of this typeclass should satisfy the following axioms:

    • Supremum associativity: ∀a, b, c∈X, sup(a, sup(b, c)) == sup(sup(a, b), c).
    • Infimum associativity: ∀a, b, c∈X, inf(a, inf(b, c)) == inf(inf(a, b), c).
    • Supremum commutativity: ∀a, b∈X, sup(a, b) == sup(b, a).
    • Infimum commutativity: ∀a, b∈X, inf(a, b) == inf(b, a).
    • Absorption of supremum w.r.t. infimum: ∀a, b∈X, sup(a, inf(a, b)) == a.
    • Absorption of infimum w.r.t. supremum: ∀a, b∈X, inf(a, sup(a, b)) == a.
55.  trait LowerSemilattice [X] extends AnyRef

    Represents a lower-semilattice, i.e., a partially-ordered set in which every finite non-empty subset has a greatest lower bound.

    Represents a lower-semilattice, i.e., a partially-ordered set in which every finite non-empty subset has a greatest lower bound.

    An instance of this typeclass should satisfy the following axioms:

    • Infimum associativity: ∀a, b, c∈X, inf(a, inf(b, c)) == inf(inf(a, b), c).
    • Infimum commutativity: ∀a, b∈X, inf(a, b) == inf(b, a).
    • Infimum idempotency: ∀a∈X, inf(a, a) == a.
56.  trait Magma [X] extends AnyRef

    Represents a magma, which is a set endowed with a binary operator.

    Represents a magma, which is a set endowed with a binary operator. No laws is specified with regards to the operator (except closure, which is satisfied by the type system).

    Since

    0.4.0

57.  trait MetricSpace [-X, +S] extends AnyRef

58.  trait Module [X, S] extends MultiplicativeAction[X, S] with AdditiveCGroup[X]

    Represents a module over a ring.

    Represents a module over a ring.

    A module over a ring is an additive Abelian group (poly.algebra.AdditiveCGroup) on vectors that supports a linear distributive scaling function. It is a generalization of a vector space because the scalars need only be a ring (poly.algebra.Ring).

    An instance of this typeclass should satisfy the following axioms:

    • S is a ring.
    • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
    • Additive identity: ∀a∈X, a + 0 == 0 + a == a.
    • Additive invertibility: \(\forall a\in X, \exists -a \in X, a + (-a) == (-a) + a == 0\).
    • Additive commutativity: ∀a, b∈X, a + b == b + a.
    • Compatibility: \(\forall k, l \in S, \forall a \in X, k(la) = (kl)a \).
    • Scaling identity: \(\forall a\in X, 1a = a\).
    • Distributivity of scaling w.r.t. vector addition: \(\forall k\in S, \forall a, b\in X, k(a+b) == ka + kb \).
    • Distributivity of scaling w.r.t. scalar addition: \(\forall k, l \in S, \forall a\in X, (k+l)a = ka + la. \).
59.  trait Monoid [M] extends Semigroup[M] with HasIdentity[M]

    Represents a monoid, i.e., a semigroup with an identity element.

    Represents a monoid, i.e., a semigroup with an identity element.

    An instance of this typeclass should satisfy the following axioms:

    • Associativity: \(\forall a, b, c \in S, (ab)c = a(bc)\).
    • Identity: \( \forall a \in X, ia = ai = a\), where \(i\) is the identity element.
60.  trait MonoidAction [X, S] extends SemigroupAction[X, S]
61.  trait MultiplicativeAction [X, S] extends AnyRef
    Since

    0.2.10

62.  trait MultiplicativeCGroup [X] extends MultiplicativeGroup[X] with MultiplicativeCMonoid[X]

    Represents an multiplicative Abelian (commutative) group.

    Represents an multiplicative Abelian (commutative) group.

    An instance of this typeclass should satisfy the following axioms:

    • Multiplicative associativity: ∀a, b, c∈X, (a * b) * c == a * (b * c).
    • Multiplicative identity: ∀a∈X, a * 1 == 1 * a == a.
    • Multiplicative invertibility: ∀a∈X, ∃a^-1∈X, a * a^-1 == a^-1 * a == 1.
    • Multiplicative commutativity: ∀a, b∈X, a * b == b * a.
63.  trait MultiplicativeCMonoid [X] extends MultiplicativeMonoid[X] with MultiplicativeCSemigroup[X]

    Represents a multiplicative commutative monoid.

    Represents a multiplicative commutative monoid.

    An instance of this typeclass should satisfy the following axioms:

    • Multiplicative associativity: ∀a, b, c∈X, (a * b) * c == a * (b * c).
    • Multiplicative identity: ∀a∈X, a * 1 == 1 * a == a.
    • Multiplicative commutativity: ∀a, b∈X, a * b == b * a.
64.  trait MultiplicativeCSemigroup [X] extends MultiplicativeSemigroup[X]

    Represents a multiplicative commutative semigroup.

    Represents a multiplicative commutative semigroup.

    An instance of this typeclass should satisfy the following axioms:

    • Multiplicative associativity: ∀a, b, c∈X, (a * b) * c == a * (b * c).
    • Multiplicative commutativity: ∀a, b∈X, a * b == b * a.
65.  trait MultiplicativeGroup [X] extends MultiplicativeMonoid[X]

    Represents an multiplicative group.

    Represents an multiplicative group. A multiplicative group is a multiplicative monoid (with operation * and identity 1) that is also invertible.

    An instance of this typeclass should satisfy the following axioms:

    • Multiplicative associativity: ∀a, b, c∈X, (a * b) * c == a * (b * c).
    • Multiplicative identity: ∀a∈X, a * 1 == 1 * a == a.
    • Multiplicative invertibility: ∀a∈X, ∃a^-1∈X, a * a^-1 == a^-1 * a == 1.
66.  trait MultiplicativeGroupAction [X, S] extends MultiplicativeMonoidAction[X, S]
67.  trait MultiplicativeMonoid [X] extends MultiplicativeSemigroup[X] with HasOne[X]

    Represents a multiplicative monoid.

    Represents a multiplicative monoid. A multiplicative monoid is a multiplicative semigroup with a multiplicative identity 1.

    An instance of this typeclass should satisfy the following axioms:

    • Multiplicative associativity: ∀a, b, c∈X, (a * b) * c == a * (b * c).
    • Multiplicative identity: ∀a∈X, a * 1 == 1 * a == a.

    X

    Type of element

68.  trait MultiplicativeMonoidAction [X, S] extends MultiplicativeSemigroupAction[X, S]
69.  trait MultiplicativeSemigroup [X] extends AnyRef

    Represents a multiplicative semigroup.

    Represents a multiplicative semigroup. An multiplicative semigroup is a semigroup with the binary operation mul (*).

    An instance of this typeclass should satisfy the following axiom:

    • Multiplicative associativity: ∀a, b, c∈X, (a * b) * c == a * (b * c).
70.  trait MultiplicativeSemigroupAction [X, S] extends MultiplicativeAction[X, S]
71.  trait MultiplicativeTorsor [X, G] extends MultiplicativeGroupAction[X, G]
72.  trait NormedVectorSpace [V, F] extends VectorSpace[V, F] with MetricSpace[V, F]

    Represents a vector space equipped with a norm.

    Represents a vector space equipped with a norm.

    Since

    0.1.0

73.  trait Order [-X] extends EqLattice[X]

    Represents a weak order.

    Represents a weak order.

    A weak order is a generalization of a total order in which some elements may be tied. Weak orders appear ubiquitously in ordered data structures and sorting algorithms.

    An instance of this typeclass should satisfy the following axioms:

    • Transitivity: ∀a, b, c∈X, a <= b and b <= c implies a <= c.
    • Totality: ∀a, b∈X, a <= b or b <= a.
74.  trait OrderedAdditiveCGroup [X] extends OrderedAdditiveGroup[X] with AdditiveCGroup[X] with MetricSpace[X, X]

75.  trait OrderedAdditiveGroup [X] extends AdditiveGroup[X] with Order[X]

76.  trait OrderedField [X] extends Field[X] with OrderedRing[X]
    Since

    0.2.4

77.  trait OrderedHashing [-X] extends Order[X] with Hashing[X]

    Represents a weak order endowed with a hashing function.

    Represents a weak order endowed with a hashing function.

    Since

    0.3.3

78.  trait OrderedRing [X] extends Ring[X] with OrderedAdditiveCGroup[X] with SignOps[X]
    Since

    0.2.4

79.  trait PartialOrder [-X] extends Eq[X]

    Represents a partial order.

    Represents a partial order.

    Partial orders are a generalization of total orders in which some element pairs may not be compared.

    An instance of this typeclass should satisfy the following axioms:

    • Reflexivity: ∀a∈X, a <= a.
    • Antisymmetry: ∀a, b∈X, a <= b and b <= a implies a == b.
    • Transitivity: ∀a, b, c∈X, a <= b and b <= c implies a <= c.
80.  trait PowerOps [X] extends AnyRef
81.  trait Priority1Implicits extends Priority2Implicits
82.  trait Priority2Implicits extends AnyRef
83.  trait Ring [X] extends Semiring[X] with AdditiveCGroup[X]

    Represents a ring.

    Represents a ring. A ring is an Abelian additive group together with a multiplicative monoid.

    An instance of this typeclass should satisfy the following axioms:

    • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
    • Additive identity: ∀a∈X, a + 0 == 0 + a == a.
    • Additive invertibility: \(\forall a\in X, \exists -a \in X, a + (-a) == (-a) + a == 0\).
    • Additive commutativity: ∀a, b∈X, a + b == b + a.
    • Multiplicative associativity: ∀a, b, c∈X, (a * b) * c == a * (b * c).
    • Multiplicative identity: ∀a∈X, a * 1 == 1 * a == a.
    • Distributivity of multiplication w.r.t. addition: ∀a, b, c∈X, (a + b) * c == a * c + b * c and a * (b + c) == a * b + a * c.

    Since

    0.1.0

84.  trait Semigroup [S] extends Magma[S]

    Typeclass for semigroups.

    Typeclass for semigroups. A semigroup is a set equipped with an associative binary operation.

    An instance of this typeclass should satisfy the following axiom:

    • Associativity: \(\forall a, b, c \in S, (ab)c = a(bc)\).
85.  trait SemigroupAction [X, S] extends Action[X, S]
86.  trait Semiring [X] extends AdditiveCMonoid[X] with MultiplicativeMonoid[X]

    Represents a semiring.

    Represents a semiring. A semiring is a commutative additive monoid as well as a multiplicative monoid.

    An instance of this typeclass should satisfy the following axioms:

    • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
    • Additive identity: ∀a∈X, a + 0 == 0 + a == a.
    • Additive commutativity: ∀a, b∈X, a + b == b + a.
    • Multiplicative associativity: ∀a, b, c∈X, (a * b) * c == a * (b * c).
    • Multiplicative identity: ∀a∈X, a * 1 == 1 * a == a.
    • Distributivity of multiplication w.r.t. addition: ∀a, b, c∈X, (a + b) * c == a * c + b * c and a * (b + c) == a * b + a * c.
    • Zero as an annihilator: ∀a∈X, 0 * a == a * 0 == 0.
87.  trait SequentialOrder [X] extends Order[X]

    Represents a total order in which each element's predecessor and successor can be accessed.

    Represents a total order in which each element's predecessor and successor can be accessed. This is equivalent to the Haskell typeclass Enum.

    Since

    0.2.19

88.  trait SignOps [X] extends AnyRef

89.  trait Similarity [-X, S] extends AnyRef

    Represents a similarity measure.

90.  trait Torsor [X, G] extends GroupAction[X, G]

    Represents a torsor, also known as a principle homogeneous space of a group.

    Represents a torsor, also known as a principle homogeneous space of a group.

    Since

    0.3.13

91.  trait TrigExpOps [X] extends AnyRef

    Represents the support for trigonometric and exponential/logarithmic functions.

    Represents the support for trigonometric and exponential/logarithmic functions.

    Since

    0.2.0

92.  trait UpperSemilattice [X] extends AnyRef

    Represents a upper-semilattice, i.e., a partially-ordered set in which every finite non-empty subset has a least upper bound.

    Represents a upper-semilattice, i.e., a partially-ordered set in which every finite non-empty subset has a least upper bound.

    An instance of this typeclass should satisfy the following axioms:

    • Supremum associativity: ∀a, b, c∈X, sup(a, sup(b, c)) == sup(sup(a, b), c).
    • Supremum commutativity: ∀a, b∈X, sup(a, b) == sup(b, a).
    • Supremum idempotency: ∀a∈X, sup(a, a) == a.
93.  trait VectorSpace [X, S] extends Module[X, S]

    Represents a vector space over a field.

    Represents a vector space over a field.

    A vector space over a field (poly.algebra.Field) is a set of vectors together with a linear distributive scaling function.

    An instance of this typeclass should satisfy the following axioms:

    • S is a field.
    • Additive associativity: \(\forall a, b, c \in X, (a+b)+c = a+(b+c)\).
    • Additive identity: ∀a∈X, a + 0 == 0 + a == a.
    • Additive invertibility: \(\forall a\in X, \exists -a \in X, a + (-a) == (-a) + a == 0\).
    • Additive commutativity: ∀a, b∈X, a + b == b + a.
    • Compatibility: \(\forall k, l \in S, \forall a \in X, k(la) = (kl)a \).
    • Scaling identity: \(\forall a\in X, 1a = a\).
    • Distributivity of scaling w.r.t. vector addition: \(\forall k\in S, \forall a, b\in X, k(a+b) == ka + kb \).
    • Distributivity of scaling w.r.t. scalar addition: \(\forall k, l \in S, \forall a\in X, (k+l)a = ka + la. \).

    Since

    0.1.0