Observables on quantum structures

An observable on a quantum structure is any σ-homomorphism of quantum structures from the Borel σ-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form (−∞, t) is sufficient to derive the whole information about the observable defined on quantum structures like σ-MV-algebras, σ-lattice effect algebras, Boolean σ-algebras, monotone σ-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and systems of functions – effect-tribes.