A Note on Hilbert Algebras and Their Related Generalized Esakia Spaces

Generalized Esakia spaces are the topological duals of bounded implicative semilattices in the duality studied by G. Bezhanishvili and R. Jansana. We study the relation between a Hilbert algebra and the generalized Esakia space dual to its free implicative semilattice extension. To establish the relation we introduce a category whose objects are a generalized Esakia space together with a family of clopen up-sets that constitutes a subalgebra of the implication fragment of the Heyting algebra of the up-sets of the generalized Esakia space.