A new categorial equivalence for Stone Algebras
Ismael Calomino, Gustavo Pelaitay
TL;DR
The paper tackles providing a categorical equivalence for Stone algebras by embedding them into the framework of Kleene algebras with intuitionistic negation (KAN-algebras) and their centered variants. It defines Stone KAN-algebras and proves that the category of Stone algebras is equivalent to the category of centered Stone KAN-algebras (SKANc) using Kalman’s construction, while also offering a Monteiro-inspired method to pass between Stone algebras and Stone KAN-algebras. Kalman’s construction K(A) from a Stone algebra A yields a centered Stone KAN-algebra, and the results show isomorphisms between the various induced structures, notably M(T) ≅ K(T lozenge). Together these constructions unify Stone algebra theory with KAN-algebra techniques and provide concrete tools for translating between algebraic and categorical perspectives; future work includes extending the framework to tense operators and weak quantifiers.
Abstract
The aim of this paper is to give a categorical equivalence for Stone algebras. We introduce the variety of Stone-Kleene algebras with intuitionistic negation, or Stone KAN-algebras for short, and explore Kalman's construction for Stone algebras. We examine the centered algebras within this new variety and prove that the category of Stone algebras is equivalent to the category of centered Stone KAN-algebras. Moreover, inspired by Monteiro's construction for Nelson algebras, we propose a method to construct a centered Stone KAN-algebra from a given Stone KAN-algebra and show the connection between Kalman's construction and Monteiro's construction.