HeytingAlgebra overview

Heyting algebras are bounded (distributive) lattices that are also equipped with an additional binary operation implies (also written as ). Heyting algebras also define a complement operation not (sometimes written as ¬a)

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 BooleanAlgebra.

A HeytingAlgebra must satisfy the following laws in addition to BoundedDistributiveLattice laws:

  • Implication:
    • a → a <-> 1
    • a ∧ (a → b) <-> a ∧ b
    • b ∧ (a → b) <-> b
    • a → (b ∧ c) <-> (a → b) ∧ (a → c)
  • Complemented
    • ¬a <-> a → 0

Added in v2.0.0


Table of contents


type classes

HeytingAlgebra (interface)

Signature

export interface HeytingAlgebra<A> extends BoundedDistributiveLattice<A> {
  readonly implies: (x: A, y: A) => A
  readonly not: (x: A) => A
}

Added in v2.0.0