Heyting algebra [SEP] named after
, the inference rule "A" → "B", "A" ⊢ "B", is sound. Equivalently a Heyting algebra is a residuated lattice whose monoid operation is ∧; yet another definition is as a posetal cartesian closed category with all finite sums. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by to formalize intuitionistic logic.
As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when "a" → "b
: is an Esakia space, called the "Esakia dual" of . Moreover, is a Heyting algebra isomorphism from onto the Heyting algebra of all clopen up-sets of . Furthermore, each Esakia space is isomorphic in Esa to the Esakia dual of some Heyting algebra.
This representation of Heyting algebras by means of Esakia spaces is functorial and yields a dual equivalence between the categories
- HA of Heyting algebras and Heyting algebra homomorphisms
and
- Esa of Esakia spaces and Esakia morphisms.
Theorem: HA is