|
- /**
- * 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`
- *
- * @since 2.0.0
- */
- import { BoundedDistributiveLattice } from './BoundedDistributiveLattice'
- /**
- * @category model
- * @since 2.0.0
- */
- export interface HeytingAlgebra<A> extends BoundedDistributiveLattice<A> {
- readonly implies: (x: A, y: A) => A
- readonly not: (x: A) => A
- }
|
