On a Generalization of Heyting Algebras II

TL;DR

This paper extends the theory of $ abla$-algebras, generalizing Heyting algebras, by developing duality theories, algebraic characterizations, and logical systems, connecting algebraic, topological, and ring-theoretic perspectives.

Contribution

It introduces $ abla$-spaces, establishes dualities, provides algebraic and ring-theoretic representations, and develops logical systems with semantics and interpolation results.

Findings

Developed a duality theory for $ abla$-algebras.

Provided algebraic and ring-theoretic characterizations.

Established logical systems with semantics and interpolation.

Abstract

A $\nabla$-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 [3], we explored the algebraic properties of various varieties of $\nabla$-algebras, their subdirectly-irreducible and simple elements, their closure under Dedekind-MacNeille completion, and their Kripke-style representation. In this sequel, we first introduce $\nabla$-spaces as a common generalization of Priestley and Esakia spaces, through which we develop a duality theory for certain categories of $\nabla$-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 $\nabla$-algebras. Finally, we introduce several logical systems to capture different varieties of $\nabla$-algebras, offering their algebraic, Kripke, topological, and ring-theoretic semantics, and establish a deductive interpolation theorem for some of these systems.