c

algebra.lattice

BoolFromBoolRing

class BoolFromBoolRing[A] extends GenBoolFromBoolRng[A] with Bool[A]

Every Boolean ring gives rise to a Boolean algebra:

  • 0 and 1 are preserved;
  • ring multiplication (times) corresponds to and;
  • ring addition (plus) corresponds to xor;
  • a or b is then defined as a xor b xor (a and b);
  • complement (¬a) is defined as a xor 1.
Source
Bool.scala
Linear Supertypes
Bool[A], Heyting[A], BoundedDistributiveLattice[A], BoundedLattice[A], BoundedMeetSemilattice[A], GenBoolFromBoolRng[A], GenBool[A], BoundedJoinSemilattice[A], DistributiveLattice[A], Lattice[A], MeetSemilattice[A], JoinSemilattice[A], Serializable, Serializable, AnyRef, Any
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. BoolFromBoolRing
  2. Bool
  3. Heyting
  4. BoundedDistributiveLattice
  5. BoundedLattice
  6. BoundedMeetSemilattice
  7. GenBoolFromBoolRng
  8. GenBool
  9. BoundedJoinSemilattice
  10. DistributiveLattice
  11. Lattice
  12. MeetSemilattice
  13. JoinSemilattice
  14. Serializable
  15. Serializable
  16. AnyRef
  17. Any
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Visibility
  1. Public
  2. All

Instance Constructors

  1. new BoolFromBoolRing(orig: BoolRing[A])

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. def and(a: A, b: A): A
    Definition Classes
    GenBoolFromBoolRng → GenBool
  5. def asBoolRing: BoolRing[A]

    Every Boolean algebra is a BoolRing, with multiplication defined as and and addition defined as xor.

    Every Boolean algebra is a BoolRing, with multiplication defined as and and addition defined as xor. Bool does not extend BoolRing because, e.g. we might want a Bool[Int] and CommutativeRing[Int] to refer to different structures, by default.

    Note that the ring returned by this method is not an extension of the Rig returned from BoundedDistributiveLattice.asCommutativeRig.

    Definition Classes
    BoolFromBoolRing → Bool → GenBoolFromBoolRng → GenBool
  6. def asCommutativeRig: CommutativeRig[A]

    Return a CommutativeRig using join and meet.

    Return a CommutativeRig using join and meet. Note this must obey the commutative rig laws since meet(a, one) = a, and meet and join are associative, commutative and distributive.

    Definition Classes
    BoundedDistributiveLattice
  7. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  8. def clone(): AnyRef
    Attributes
    protected[lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( ... ) @native() @HotSpotIntrinsicCandidate()
  9. def complement(a: A): A
    Definition Classes
    BoolFromBoolRing → Heyting
  10. def dual: Bool[A]

    This is the lattice with meet and join swapped

    This is the lattice with meet and join swapped

    Definition Classes
    Bool → BoundedDistributiveLattice → BoundedLattice → Lattice
  11. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  12. def equals(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  13. final def getClass(): Class[_]
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  14. def hashCode(): Int
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  15. def imp(a: A, b: A): A
    Definition Classes
    Bool → Heyting
  16. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  17. def isOne(a: A)(implicit ev: Eq[A]): Boolean
    Definition Classes
    BoundedMeetSemilattice
  18. def isZero(a: A)(implicit ev: Eq[A]): Boolean
    Definition Classes
    BoundedJoinSemilattice
  19. def join(a: A, b: A): A
    Definition Classes
    BoolFromBoolRing → Heyting → GenBool → JoinSemilattice
  20. def joinPartialOrder(implicit ev: Eq[A]): PartialOrder[A]
    Definition Classes
    JoinSemilattice
  21. def joinSemilattice: BoundedSemilattice[A]
    Definition Classes
    BoundedJoinSemilattice → JoinSemilattice
  22. def meet(a: A, b: A): A
    Definition Classes
    BoolFromBoolRing → Heyting → GenBool → MeetSemilattice
  23. def meetPartialOrder(implicit ev: Eq[A]): PartialOrder[A]
    Definition Classes
    MeetSemilattice
  24. def meetSemilattice: BoundedSemilattice[A]
    Definition Classes
    BoundedMeetSemilattice → MeetSemilattice
  25. def nand(a: A, b: A): A
    Definition Classes
    Heyting
  26. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  27. def nor(a: A, b: A): A
    Definition Classes
    Heyting
  28. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  29. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  30. def nxor(a: A, b: A): A
    Definition Classes
    Heyting
  31. def one: A
    Definition Classes
    BoolFromBoolRing → BoundedMeetSemilattice
  32. def or(a: A, b: A): A
    Definition Classes
    GenBoolFromBoolRng → GenBool
  33. final def synchronized[T0](arg0: ⇒ T0): T0
    Definition Classes
    AnyRef
  34. def toString(): String
    Definition Classes
    AnyRef → Any
  35. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  36. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... ) @native()
  37. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  38. def without(a: A, b: A): A

    The operation of relative complement, symbolically often denoted a\b (the symbol for set-theoretic difference, which is the meaning of relative complement in the lattice of sets).

    The operation of relative complement, symbolically often denoted a\b (the symbol for set-theoretic difference, which is the meaning of relative complement in the lattice of sets).

    Definition Classes
    BoolFromBoolRing → Bool → GenBoolFromBoolRng → GenBool
  39. def xor(a: A, b: A): A

    Logical exclusive or, set-theoretic symmetric difference.

    Logical exclusive or, set-theoretic symmetric difference. Defined as a\b ∨ b\a.

    Definition Classes
    Bool → GenBool → Heyting
  40. def zero: A
    Definition Classes
    GenBoolFromBoolRng → BoundedJoinSemilattice

Deprecated Value Members

  1. def finalize(): Unit
    Attributes
    protected[lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( classOf[java.lang.Throwable] ) @Deprecated @deprecated
    Deprecated

    (Since version ) see corresponding Javadoc for more information.

Inherited from Bool[A]

Inherited from Heyting[A]

Inherited from BoundedDistributiveLattice[A]

Inherited from BoundedLattice[A]

Inherited from BoundedMeetSemilattice[A]

Inherited from GenBoolFromBoolRng[A]

Inherited from GenBool[A]

Inherited from BoundedJoinSemilattice[A]

Inherited from DistributiveLattice[A]

Inherited from Lattice[A]

Inherited from MeetSemilattice[A]

Inherited from JoinSemilattice[A]

Inherited from Serializable

Inherited from Serializable

Inherited from AnyRef

Inherited from Any

Ungrouped