Heyting

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

Implication obeys the following laws:

a → a = 1
a ∧ (a → b) = a ∧ b
b ∧ (a → b) = b
a → (b ∧ c) = (a → b) ∧ (a → c)

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:

a ∧ ¬a = 0

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

Abstract Value Members

abstract def and(a: A, b: A): A
abstract def complement(a: A): A
abstract def getClass(): Class[_]

Definition Classes
Any
abstract def imp(a: A, b: A): A
abstract def one: A

Definition Classes
BoundedMeetSemilattice
abstract def or(a: A, b: A): A
abstract def zero: A

Definition Classes
BoundedJoinSemilattice

Concrete Value Members

final def !=(arg0: Any): Boolean

Definition Classes
Any
final def ##(): Int

Definition Classes
Any
final def ==(arg0: Any): Boolean

Definition Classes
Any
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
final def asInstanceOf[T0]: T0

Definition Classes
Any
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
def equals(arg0: Any): Boolean

Definition Classes
Any
def hashCode(): Int

Definition Classes
Any
final def isInstanceOf[T0]: Boolean

Definition Classes
Any
def isOne(a: A)(implicit ev: Eq[A]): Boolean

Definition Classes
BoundedMeetSemilattice
def isZero(a: A)(implicit ev: Eq[A]): Boolean

Definition Classes
BoundedJoinSemilattice
def join(a: A, b: A): A

Definition Classes
Heyting → JoinSemilattice
def joinPartialOrder(implicit ev: Eq[A]): PartialOrder[A]

Definition Classes
JoinSemilattice
def joinSemilattice: BoundedSemilattice[A]

Definition Classes
BoundedJoinSemilattice → JoinSemilattice
def meet(a: A, b: A): A

Definition Classes
Heyting → MeetSemilattice
def meetPartialOrder(implicit ev: Eq[A]): PartialOrder[A]

Definition Classes
MeetSemilattice
def meetSemilattice: BoundedSemilattice[A]

Definition Classes
BoundedMeetSemilattice → MeetSemilattice
def nand(a: A, b: A): A
def nor(a: A, b: A): A
def nxor(a: A, b: A): A
def toString(): String

Definition Classes
Any
def xor(a: A, b: A): A

Related Docs: object Heyting | package lattice

trait Heyting[A] extends BoundedDistributiveLattice[A]

Abstract Value Members

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

abstract def complement(a: A): A

abstract def getClass(): Class[_]

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

abstract def one: A

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

abstract def zero: A

Concrete Value Members

final def !=(arg0: Any): Boolean

final def ##(): Int

final def ==(arg0: Any): Boolean

def asCommutativeRig: CommutativeRig[A]

final def asInstanceOf[T0]: T0

def dual: BoundedDistributiveLattice[A]

def equals(arg0: Any): Boolean

def hashCode(): Int

final def isInstanceOf[T0]: Boolean

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

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

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

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

def joinSemilattice: BoundedSemilattice[A]

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

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

def meetSemilattice: BoundedSemilattice[A]

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

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

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

def toString(): String

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