basis.math

OrderedField

trait OrderedField extends OrderedRing with Field

A totally ordered abstract field structure. Addition associates and commutes, and multiplication associates, commutes, and distributes over addition. Addition and multiplication both have an identity element, every every element has an additive inverse, and every element except zero has a multiplicative inverse. To the extent practicable, the following axioms should hold.

Axioms for addition:

Axioms for multiplication:

The distributive law:

Order axioms:

Source
OrderedField.scala
Example: 

  1. // You can abstract over ordered fields by parameterizing a class or
// function with a subtype of OrderedField with Singleton. Type elements
// with the #Element type projection of your OrderedField type parameter.
def testOrderedFieldOperations[F <: OrderedField with Singleton](a: F#Element, b: F#Element, c: F#Element): Unit = {
  assert(a + b == b + a, "commutativity of addition")
  assert((a + b) + c == a + (b + c), "associativity of addition")
  assert(a * b == b * a, "commutativity of multiplication")
  assert((a * b) * c == a * (b * c), "associativity of multiplication")
  assert(a * (b + c) == (a * b) + (a * c), "distributivity of multiplication over addition")
  if (a <= b) assert((a min b) == a, "existence of minima")
  if (a <= b) assert((a max b) == b, "existence of maxima")
}

// Alternatively, functions can use path-dependent types of an OrderedField parameter.
def testOrderedFieldIdentities(F: OrderedField)(a: F.Element, b: F.Element): Unit = {
  import F._
  assert(zero + a == a, "existence of additive identity")
  assert(a + (-a) == zero, "existence of additive inverse")
  assert(unit != zero && unit * a == a, "existence of multiplicative identity")
  assert(a * a.inverse == unit, "existence of multiplicative inverse")
  if (a <= b && b <= a) assert(a == b, "antisymmetry of ordering")
  if (a <= b && b <= c) assert(a <= c, "transitivity of ordering")
  assert(a <= b || b <= a, "totality of ordering")
}
Version

0.1

Since

0.0

Linear Supertypes
Field, OrderedRing, Ring, AnyRef, Any
Known Subclasses
Real, RealField
Type Hierarchy Learn more about scaladoc diagrams
Ordering
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Inherited
  1. OrderedField
  2. Field
  3. OrderedRing
  4. Ring
  5. AnyRef
  6. Any
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Visibility
  1. Public
  2. All

Type Members

  1. abstract type Element <: OrderedFieldElement

    The type of elements in this ordered field.

    The type of elements in this ordered field.

    Definition Classes
    OrderedField → Field → OrderedRing → Ring

  2. trait FieldElement extends RingElement

    Definition Classes
    Field

  3. trait OrderedFieldElement extends OrderedRingElement with FieldElement

  4. trait OrderedRingElement extends RingElement

    Definition Classes
    OrderedRing

  5. trait RingElement extends Any

    An element in this ring.

    An element in this ring.

    Definition Classes
    Ring

Abstract Value Members

  1. abstract def unit: Element

    Returns the multiplicative identity of this ordered field.

    Returns the multiplicative identity of this ordered field.

    Definition Classes
    OrderedField → Field → OrderedRing → Ring

  2. abstract def zero: Element

    Returns the additive identity of this ordered field.

    Returns the additive identity of this ordered field.

    Definition Classes
    OrderedField → Field → OrderedRing → Ring

Concrete Value Members

  1. final def !=(arg0: Any): Boolean

    Definition Classes
    AnyRef → Any

  2. final def ##(): Int

    Definition Classes
    AnyRef → Any

  3. final def ==(arg0: Any): Boolean

    Definition Classes
    AnyRef → Any

  4. final def asInstanceOf[T0]: T0

    Definition Classes
    Any

  5. def clone(): AnyRef

    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

  6. final def eq(arg0: AnyRef): Boolean

    Definition Classes
    AnyRef

  7. def equals(arg0: Any): Boolean

    Definition Classes
    AnyRef → Any

  8. def finalize(): Unit

    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( classOf[java.lang.Throwable] )

  9. final def getClass(): Class[_]

    Definition Classes
    AnyRef → Any

  10. def hashCode(): Int

    Definition Classes
    AnyRef → Any

  11. final def isInstanceOf[T0]: Boolean

    Definition Classes
    Any

  12. final def ne(arg0: AnyRef): Boolean

    Definition Classes
    AnyRef

  13. final def notify(): Unit

    Definition Classes
    AnyRef

  14. final def notifyAll(): Unit

    Definition Classes
    AnyRef

  15. final def synchronized[T0](arg0: ⇒ T0): T0

    Definition Classes
    AnyRef

  16. def toString(): String

    Definition Classes
    AnyRef → Any

  17. final def wait(): Unit

    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

  18. final def wait(arg0: Long, arg1: Int): Unit

    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

  19. final def wait(arg0: Long): Unit

    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

Inherited from Field

Inherited from OrderedRing

Inherited from Ring

Inherited from AnyRef

Inherited from Any

Ungrouped