States on Quantum Structures Versus Integrals

Abstract

We show conditions when a state on a quantum structure E like an effect algebra, a pseudo effect algebra E satisfying some kind of the Riesz Decomposition Properties (RDP) or on an MV-algebra, a BL-algebra, a pseudo MV-algebra and a pseudo BL-algebra is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of (RDP), the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K. The same is true for states on an MV-algebra and a BL-algebra and their noncommutative variants.