On a Generalization of Heyting Algebras II

A ∇ ∇ -algebra is a natural generalization of a Heyting algebra, unifying several algebraic structures, including bounded lattices, Heyting algebras, temporal Heyting algebras, and the algebraic representation of dynamic topological systems. In the prequel to this paper [1], we explored the algebraic properties of various varieties of ∇ ∇ -algebras, their subdirectly-irreducible and simple elements, their closure under Dedekind-MacNeille completion, and their Kripke-style representation. In this sequel, we first introduce ∇ ∇ -spaces as a common generalization of Priestley and Esakia spaces, through which we develop a duality theory for certain categories of ∇ ∇ -algebras. Then, we reframe these dualities in terms of spectral spaces and provide an algebraic characterization of natural families of dynamic topological systems over Priestley, Esakia, and spectral spaces. Additionally, we present a ring-theoretic representation for some families of ∇ ∇ -algebras.