Loomis-Sikorski theorem and Stone duality for effect algebras with internal state

Abstract

Recently Flaminio and Montagna, [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 σ-complete effect algebras with inter- nal state. In addition, we show that the category of divisible state-morphism effect algebras satisfying (RDP) and countable interpolation with an order de- termining system of states is dual to the category of Bauer simplices Ω such that ∂eΩ is an F-space.