Esakia duality

Esakia duality

In mathematics, Esakia duality is the dual equivalence between the category of Heyting algebras and the category of Esakia spaces. Esakia duality provides an order-topological representation of Heyting algebras via Esakia spaces.

Let Esa denote the category of Esakia spaces and Esakia morphisms.

Let H be a Heyting algebra, X denote the set of prime filters of H, and ≤ denote set-theoretic inclusion on the prime filters of H. Also, for each a∈ H, let φ(a) = {x∈ X : a∈ x} , and let τ denote the topology on X generated by {φ(a), X−φ(a) : a∈ H}.

Theorem: (X,τ,≤) is an Esakia space, called the Esakia dual of H. Moreover, φ is a Heyting algebra isomorphism from H onto the Heyting algebra of all clopen up-sets of (X,τ,≤). 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 category HA of Heyting algebras and Heyting algebra homomorphisms and the category Esa of Esakia spaces and Esakia morphisms.

Theorem: HA is dually equivalent to Esa.

References

• Esakia, L. (1974). Topological Kripke models. Soviet Math. Dokl., 15 147--151.

• Esakia, L. (1985). Heyting Algebras I. Duality Theory (Russian). Metsniereba, Tbilisi.

• Bezhanishvili, N. (2006). Lattices of Intermediate and Cylindric Modal Logics. ILLC, University of Amsterdam.

See Also

• Duality theory for distributive lattices