Stone and double Stone algebras: Boolean and Rough Set Representations, 3-valued and 4-valued Logics

TL;DR

This paper provides structural representations of Stone, dual Stone, and double Stone algebras using ordered tuples of sets, and develops 3- and 4-valued logical semantics with soundness and completeness proofs.

Contribution

It extends Moisil's algebraic models to Stone and double Stone algebras, introducing set-based representations and multi-valued logical semantics.

Findings

Structural representations of Stone and double Stone algebras as ordered tuples of sets.

3-valued semantics for Stone and dual Stone algebras.

4-valued semantics for double Stone algebras with soundness and completeness.

Abstract

Moisil in 1941, while constructing the algebraic models of n-valued {\L}ukasiewicz logic defined the set $B^{[n]}$,where $B$ is a Boolean algebra and `n' being a natural number. Further it was proved by Moisil himself the representations of n-valued {\L}ukasiewicz Moisil algebra in terms of $B^{[n]}$. In this article, structural representation results for Stone, dual Stone and double Stone algebras are proved similar to Moisil's work by showing that elements of these algebras can be looked upon as monotone ordered tuple of sets. 3-valued semantics of logic for Stone algebra, dual Stone algebras and 4-valued semantics of logic for double Stone algebras are proposed and established soundness and completeness results.