Representation of States on Effect-Tribes and Effect Algebras by Integrals

We describe $\sigma$-additive states on effect-tribes by integrals. Effect-tribes are monotone $\sigma$-complete effect algebras of functions where operations are defined by points. Then we show that every state on an effect algebra is an integral through a Borel regular probability measure. Finally, we show that every $\sigma$-convex combination of extremal states on a monotone $\sigma$-complete effect algebra is a Jauch-Piron state.Comment: 20 page

Smearing of Observables and Spectral Measures on Quantum Structures

An observable on a quantum structure is any $\sigma$-homomorphism of quantum structures from the Borel $\sigma$-algebra of the real line into the quantum structure which is in our case a monotone $\sigma$-complete effect algebras with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean $\sigma$-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there is its spectral measure

Toward a probability theory for product logic: states, integral representation and reasoning

The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure. Furthermore, the relation between states and measures is shown to be one-one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the measure is a state in the above sense.Comment: 27 pages, 1 figur

States on pseudo effect algebras and integrals

We show that every state on an interval pseudo effect algebra $E$ satisfying some kind of the Riesz Decomposition Properties (RDP) is an integral through a regular Borel probability measure defined on the Borel $\sigma$-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.