basis.math

OrderedRing

trait OrderedRing extends Ring

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

Axioms for addition:

Axioms for multiplication:

The distributive law:

Order axioms:

Source
OrderedRing.scala
Example: 

  1. // You can abstract over ordered rings by parameterizing a class or
// function with a subtype of OrderedRing with Singleton. Type elements
// with the #Element type projection of your OrderedRing type parameter.
def testOrderedRingOperations[R <: OrderedRing with Singleton](a: R#Element, b: R#Element, c: R#Element): Unit = {
  assert(a + b == b + a, "commutativity of addition")
  assert((a + b) + c == a + (b + c), "associativity of addition")
  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 OrderedRing parameter.
def testOrderedRingIdentities(R: OrderedRing)(a: R.Element, b: R.Element): Unit = {
  import R._
  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")
  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
Ring, AnyRef, Any
Known Subclasses
Integer, OrderedField, Real, RealField
Type Hierarchy Learn more about scaladoc diagrams
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  1. OrderedRing
  2. Ring
  3. AnyRef
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  1. Public
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Type Members

  1. abstract type Element <: OrderedRingElement

    The type of elements in this ordered ring.

    The type of elements in this ordered ring.

    Definition Classes
    OrderedRing → Ring

  2. trait OrderedRingElement extends RingElement

  3. 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 ring.

    Returns the multiplicative identity of this ordered ring.

    Definition Classes
    OrderedRing → Ring

  2. abstract def zero: Element

    Returns the additive identity of this ordered ring.

    Returns the additive identity of this ordered ring.

    Definition Classes
    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 Ring

Inherited from AnyRef

Inherited from Any

Ungrouped