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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 = {x ∈ X : a ∈ x}, and let τ denote the topology on X generated by {φ, X − φ : a ∈ H}.
Theorem: 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. Furthermore, each Esakia space is isomorphic in Esa to the Esakia dual of some Heyting algebra.