Heyting algebra

In mathematics, Heyting algebras are a generalization of Boolean algebras. Heyting algebras model intuitionistic logic, in which the law of excluded middle does not in general hold.

Formally, a Heyting algebra is a bounded lattice L such that for all a and b in L there is a greatest element x of L such that a ∧ x ≤ b. This element is called the relative pseudo-complement of a with respect to b, and is denoted a⇒b (or a→b). We write ¬a for a⇒0. Heyting algebras are always distributive; this is sometimes stated as an axiom, but in fact it follows from the existence of relative pseudo-complements.

Boolean algebras are those Heyting algebras in which x = ¬¬x for all x, or, equivalently, in which x ∨ ¬x = 1 for all x. In this case, the element a⇒b is equal to ¬a ∨ b.

Every topology is also a Heyting algebra. In this case, the element A⇒B is the interior of A^c∪B, where A^c denotes the complement of the open set A.