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MV-observables and MV-algebras. (English) Zbl 0992.03081
Let \({\mathcal B}(R)\) be the \(\sigma\)-algebra of Borel subsets of \(R\), \({\mathcal T}_{\mathcal B}(R)\) the MV-algebra of all Borel measurable functions \(f: R\to [0,1]\). Usually an observable in an MV-algebra \(M\) is defined as a mapping \(x\) from the Borel \(\sigma\)-algebra \({\mathcal B}(R)\) to \(M\) such that \(x(R)\) is the greatest element of \(M\), \(x\) is additive (\(A\cap B)= \emptyset\Rightarrow x(A\cup B)= x(A)\oplus x(B)\)) and continuous from below. The author defines an MV-observable as an MV-\(\sigma\)-homomorphism from \({\mathcal T}_{\mathcal B}(R)\) to \(M\). The theory completely corresponds to category theory, of course (as is proved in the paper) it is applicable only for weakly divisible MV-algebras. A representation is presented as well as a calculus of MV-observables, which enables to construct a.e. the sum or product of MV-observables.

03G12 Quantum logic
06D35 MV-algebras
Full Text: DOI
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