Stone Duality Type Theorems for MV-Algebras with Internal State

Recently the language of MV-algebras was extended by adding a unary operation, an internal operator, called also a state-operator. A stronger version of state MV-algebras, called state-morphism MV-algebras, was given. In this article, 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.