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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 $(-\infty,t)$ is sufficient to determine uniquely the whole observable defined on quantum structures like $σ$-MV-algebras, $σ$-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 a system of functions, and an effect-tribe.