Loomis--Sikorski Theorem and Stone Duality for Effect Algebras with Internal State

TL;DR

This paper extends the Loomis--Sikorski Theorem and Stone duality to effect algebras with internal states, broadening the algebraic and topological understanding of these structures.

Contribution

It introduces the concept of a state-operator for effect algebras and generalizes key duality theorems to this broader setting.

Findings

Generalized Loomis--Sikorski Theorem for effect algebras with internal state.

Established duality between certain effect algebra categories and Bauer simplices.

Characterized effect algebras satisfying (RDP) with state systems as dual to specific F-spaces.

Abstract

Recently Flaminio and Montagna, \cite{FlMo}, extended the language of MV-algebras by adding a unary operation, called a state-operator. This notion is introduced here also for effect algebras. Having it, we generalize the Loomis--Sikorski Theorem for monotone $\sigma$-complete effect algebras with internal state. In addition, we show that the category of divisible state-morphism effect algebras satisfying (RDP) and countable interpolation with an order determining system of states is dual to the category of Bauer simplices $\Omega$ such that $\partial_e \Omega$ is an F-space.