A new categorial equivalence for Stone Algebras

TL;DR

This paper establishes a categorical equivalence for Stone algebras by introducing Stone-Kleene algebras with intuitionistic negation and exploring their constructions, connecting different algebraic frameworks.

Contribution

It introduces Stone-Kleene algebras with intuitionistic negation and proves the equivalence between the categories of Stone algebras and centered Stone KAN-algebras.

Findings

Category of Stone algebras is equivalent to centered Stone KAN-algebras

Constructed a method to derive centered Stone KAN-algebras from Stone KAN-algebras

Connected Kalman's and Monteiro's constructions for these algebras

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.