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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.

##### MSC:
 03G12 Quantum logic 06D35 MV-algebras
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