Esakia space

In mathematics, Esakia spaces are special ordered topological spaces introduced and studied by Leo Esakia in 1974.^[1] Esakia spaces play a fundamental role in the study of Heyting algebras, primarily by virtue of the Esakia duality --- the dual equivalence between the category of Heyting algebras and the category of Esakia spaces.

Definition

For a partially ordered set (X,≤) and for x∈ X, let ↓x = {y∈ X : y≤ x} and let ↑x = {y∈ X : x≤ y} . Also, for A⊆ X, let ↓A = {y∈ X : y ≤ x for some x∈ A} and ↑A = {y∈ X : y≥ x for some x∈ A} .

An Esakia space is a Priestley space (X,τ,≤) such that for each clopen subset C of the topological space (X,τ), the set ↓C is also clopen.

Equivalent definitions

There are several equivalent ways to define Esakia spaces.

Theorem:^[2] The following conditions are equivalent:

(i) (X,τ,≤) is an Esakia space.

(ii) ↑x is closed for each x∈ X and ↓C is clopen for each clopen C⊆ X.

(iii) ↓x is closed for each x∈ X and ↑cl(A) = cl(↑A) for each A⊆ X (where cl denotes the closure in X).

(iv) ↓x is closed for each x∈ X, the least closed set containing an up-set is an up-set, and the least up-set containing a closed set is closed.

Esakia morphisms

Let (X,≤) and (Y,≤) be partially ordered sets and let f : X → Y be an order-preserving map. The map f is a bounded morphism (also known as p-morphism) if for each x∈ X and y∈ Y, if f(x)≤ y, then there exists z∈ X such that x≤ z and f(z) = y.

Theorem:^[3] The following conditions are equivalent:

(1) f is a bounded morphism.

(2) f(↑x) = ↑f(x) for each x∈ X.

(3) f⁻¹(↓y) = ↓f⁻¹(y) for each y∈ Y.

Let (X, τ, ≤) and (Y, τ′, ≤) be Esakia spaces and let f : X → Y be a map. The map f is called an Esakia morphism if f is a continuous bounded morphism.

Notes

References

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

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