algebra.lattice

Heyting

trait Heyting[A] extends BoundedDistributiveLattice[A]

Heyting algebras are bounded lattices that are also equipped with an additional binary operation imp (for impliciation, also written as →).

Implication obeys the following laws:

In heyting algebras, and is equivalent to meet and or is equivalent to join; both methods are available.

Heyting algebra also define complement operation (sometimes written as ¬a). The complement of a is equivalent to (a → 0), and the following laws hold:

However, in Heyting algebras this operation is only a pseudo-complement, since Heyting algebras do not necessarily provide the law of the excluded middle. This means that there is no guarantee that (a ∨ ¬a) = 1.

Heyting algebras model intuitionistic logic. For a model of classical logic, see the boolean algebra type class implemented as Bool.

Self Type
Heyting[A]
Linear Supertypes
BoundedDistributiveLattice[A], DistributiveLattice[A], BoundedLattice[A], BoundedJoinSemilattice[A], BoundedMeetSemilattice[A], Lattice[A], MeetSemilattice[A], JoinSemilattice[A], Serializable, Serializable, Any
Known Subclasses
Bool, BoolFromBoolRing, BooleanAlgebra, DualBool
Ordering
  1. Alphabetic
  2. By inheritance
Inherited
  1. Heyting
  2. BoundedDistributiveLattice
  3. DistributiveLattice
  4. BoundedLattice
  5. BoundedJoinSemilattice
  6. BoundedMeetSemilattice
  7. Lattice
  8. MeetSemilattice
  9. JoinSemilattice
  10. Serializable
  11. Serializable
  12. Any
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  1. Public
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Abstract Value Members

  1. abstract def and(a: A, b: A): A

  2. abstract def complement(a: A): A

  3. abstract def getClass(): Class[_]

    Definition Classes
    Any

  4. abstract def imp(a: A, b: A): A

  5. abstract def one: A

    Definition Classes
    BoundedMeetSemilattice

  6. abstract def or(a: A, b: A): A

  7. abstract def zero: A

    Definition Classes
    BoundedJoinSemilattice

Concrete Value Members

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

    Definition Classes
    Any

  2. final def ##(): Int

    Definition Classes
    Any

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

    Definition Classes
    Any

  4. 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

  5. final def asInstanceOf[T0]: T0

    Definition Classes
    Any

  6. def dual: BoundedDistributiveLattice[A]

    This is the lattice with meet and join swapped

    This is the lattice with meet and join swapped

    Definition Classes
    BoundedDistributiveLattice → BoundedLattice → Lattice

  7. def equals(arg0: Any): Boolean

    Definition Classes
    Any

  8. def hashCode(): Int

    Definition Classes
    Any

  9. final def isInstanceOf[T0]: Boolean

    Definition Classes
    Any

  10. def isOne(a: A)(implicit ev: Eq[A]): Boolean

    Definition Classes
    BoundedMeetSemilattice

  11. def isZero(a: A)(implicit ev: Eq[A]): Boolean

    Definition Classes
    BoundedJoinSemilattice

  12. def join(a: A, b: A): A

    Definition Classes
    Heyting → JoinSemilattice

  13. def joinPartialOrder(implicit ev: Eq[A]): PartialOrder[A]

    Definition Classes
    JoinSemilattice

  14. def joinSemilattice: BoundedSemilattice[A]

    Definition Classes
    BoundedJoinSemilattice → JoinSemilattice

  15. def meet(a: A, b: A): A

    Definition Classes
    Heyting → MeetSemilattice

  16. def meetPartialOrder(implicit ev: Eq[A]): PartialOrder[A]

    Definition Classes
    MeetSemilattice

  17. def meetSemilattice: BoundedSemilattice[A]

    Definition Classes
    BoundedMeetSemilattice → MeetSemilattice

  18. def nand(a: A, b: A): A

  19. def nor(a: A, b: A): A

  20. def nxor(a: A, b: A): A

  21. def toString(): String

    Definition Classes
    Any

  22. def xor(a: A, b: A): A

Inherited from BoundedDistributiveLattice[A]

Inherited from DistributiveLattice[A]

Inherited from BoundedLattice[A]

Inherited from BoundedJoinSemilattice[A]

Inherited from BoundedMeetSemilattice[A]

Inherited from Lattice[A]

Inherited from MeetSemilattice[A]

Inherited from JoinSemilattice[A]

Inherited from Serializable

Inherited from Serializable

Inherited from Any

Ungrouped