* Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix " ' ", and governed by the axioms x ' x = 1, x ( x ' y )

* Heyting algebra

( See the section titled Heyting algebra semantics below for a sort of " infinitely-many valued logic " interpretation of intuitionistic logic.

* A Heyting algebra is a Cartesian closed ( bounded ) lattice.

The structure on its sub-object classifier is that of a Heyting algebra.

Heyting algebra -- Higher-order predicate -- Horn clause -- Hypothetical syllogism

In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice ( with join and meet operations written ∧ and ∨ and with least element 0 and greatest element 1 ) equipped with a binary operation a → b of implication such that ( a → b )∧ a ≤ b, and moreover a → b is the greatest such in the sense that if c ∧ a ≤ b then c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound.

Equivalently a Heyting algebra is a residuated lattice whose monoid operation a • b is a ∧ b ; yet another definition is as a posetal cartesian closed category with all finite sums.

Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.

In the finite case every nonempty distributive lattice, in particular every nonempty finite chain, is automatically bounded and complete and hence a Heyting algebra.

It can further be shown that a ≤ ¬¬ a, although the converse, ¬¬ a ≤ a, is not true in general, that is, double negation does not hold in general in a Heyting algebra.

Heyting algebras generalize Boolean algebras in the sense that a Heyting algebra satisfying a ∨¬ a = 1 ( excluded middle ), equivalently ¬¬ a = a ( double negation ), is a Boolean algebra.

Those elements of a Heyting algebra of the form ¬ a comprise a Boolean lattice, but in general this is not a subalgebra of H ( see below ).

The internal logic of an elementary topos is based on the Heyting algebra of subobjects of the terminal object 1 ordered by inclusion, equivalently the morphisms from 1 to the subobject classifier Ω.

Every Heyting algebra with exactly one coatom is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new top.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

For instance, in Heyting arithmetic, one can prove that for any proposition p which does not contain quantifiers, is a theorem ( where x, y, z ... are the free variables in the proposition p ).

In intuitionistic logic, according to the Brouwer – Heyting – Kolmogorov interpretation, the negation of a proposition p is the proposition whose proofs are the refutations of p. In Kripke semantics where the semantic values of formulae are sets of possible worlds, negation is set-theoretic complementation.

A bounded lattice is a Heyting algebra if and only if all mappings are the lower adjoint of a monotone Galois connection.

Given a bounded lattice with largest and smallest elements 1 and 0, and a binary operation, these together form a Heyting algebra if and only if the following hold:

* Every totally ordered set that is a bounded lattice is also a Heyting algebra, where is equal to when, and 1 otherwise.

A complete Heyting algebra is a Heyting algebra that is a complete lattice.

The free object | free Heyting algebra over one generator ( aka Rieger – Nishimura lattice )

* Every Heyting algebra is a distributive lattice.

Hence Ω ( X ) is not an arbitrary complete lattice but a complete Heyting algebra ( also called frame or locale-the various names are primarily used to distinguish several categories that have the same class of objects but different morphisms: frame morphisms, locale morphisms and homomorphisms of complete Heyting algebras ).

A Heyting algebra that is a complete lattice is called a complete Heyting algebra.

Details on this characterization can be found in the articles on the " lattice-like " structures for which this is typically considered: see semilattice, lattice, Heyting algebra, and Boolean algebra.

Typical examples of distributive lattice are totally ordered sets, Boolean algebras, and Heyting algebras.

Constructive proofs can be seen as defining certified mathematical algorithms: this idea is explored in the Brouwer – Heyting – Kolmogorov interpretation of constructive logic, the Curry – Howard correspondence between proofs and programs, and such logical systems as Per Martin-Löf's Intuitionistic Type Theory, and Thierry Coquand and Gérard Huet's Calculus of Constructions.

Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism.

Given a set with three binary operations and, and two distinguished elements 0 and 1, then is a Heyting algebra for these operations ( and the relation defined by the condition that when ) if and only if the following conditions hold for any elements and of:

* P is a Heyting algebra, i. e. the operation has a right adjoint ( also called the lower adjoint of a ( monotone ) Galois connection ), for each element x of P.

Heyting algebras and interior algebras are the Lindenbaum – Tarski algebras for intuitionistic logic and the modal logic S4, respectively.

In fact, since "=" is the only predicate symbol in Heyting arithmetic, it then follows that, for any quantifier-free formula p, is a theorem ( where x, y, z … are the free variables in p ).

Hence no finite set of finite Heyting algebras can supply all the counterexamples to non-laws of Heyting algebra.

Nevertheless it is decidable whether an equation holds of all Heyting algebras.

The collection of all open subsets of a topological space X forms a complete Heyting algebra.

The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra.

Peter Freyd observed that it is a ( meta ) theorem that in free Heyting algebras and free topoi ( and all sorts of variations thereon ) that the disjunction property holds.

It follows that even among the finite Heyting algebras there exist infinitely many that are subdirectly irreducible, no two of which have the same equational theory.

In any Heyting algebra, one defines the pseudo-complement of any element by setting.