Stone Duality Type Theorems for MV-algebras with Internal State

TL;DR

This paper establishes Stone duality theorems connecting categories of MV-algebras with internal states to topological spaces, extending duality theory to new algebraic structures with state-operators.

Contribution

It introduces duality theorems for categories of MV-algebras with internal states and their topological counterparts, expanding the scope of Stone duality.

Findings

Duality between Boolean algebras with state-operators and compact Hausdorff spaces with idempotent functions

Duality between weakly divisible σ-complete state-morphism MV-algebras and Bauer simplices

Characterization of these dualities in terms of topological and algebraic structures

Abstract

Recently in \cite{FM, FlMo}, the language of MV-algebras was extended by adding a unary operation, an internal operator, called also a state-operator. In \cite{DD1}, a stronger version of state MV-algebras, called state-morphism MV-algebras was given. In this paper, we present Stone Duality Theorems for (i) the category of Boolean algebras with a fixed state-operator and the category of compact Hausdorff topological spaces with a fixed idempotent continuous function, and for (ii) the category of weakly divisible $\sigma$-complete state-morphism MV-algebras and the category of Bauer simplices whose set of extreme points is basically disconnected and with a fixed idempotent continuous function.