On States on MV-algebras and their Applications

1 Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia E-mail: [email protected] In the frames of quantum structures, states are a very important notion that model probability on an algebraic structure. Already G. Boole observed that to calculate probability of event of the structure M , it is important to know only which two events A and B we can add such that if C = A+B, and P is a probability, then P (A+B) = P (A) + P (B); and the operation + is a partial one on M. If M is a Boolean algebra, then A+B := A∪B whenever A∩B = ∅ or equivalently, A ≤ B′. Such two events A and B are said to be mutually excluding or summable, [DvPu]. Therefore, the state or finitely additive state on an algebraic structure (M ; +,′ , 0, 1) is any mapping s : M → [0, 1] such that (i) s(1) = 1, and (ii) s(a + b) = s(a) + s(b) whenever a + b is defined in M. The unary operation ′ is an orthogonal complement or a negation. The basic task is how to define a state on an algebraic structure like (M ;⊕, ,∗ , 0, 1) or (M ;⊕, ,− ,∼ , 0, 1), that is, how to derive a partial operation + from the ⊕, ? In more complicated structures, like orthomodular lattices and effect algebras, the most important example is the system L(H) of all closed subspaces of a Hilbert space H or the system of all Hermitian operators E(H) that are between the zero operator and the identity operator. Here the σ-additive states are of the form