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The spectrum of the sum of observables on \(\sigma\)-complete MV-effect algebras. (English) Zbl 1415.06004
Summary: The natural question about the sum of observables on \(\sigma\)-complete MV-effect algebras, which was recently defined by A. Dvurečenskij [Int. J. Theor. Phys. 57, No. 3, 637–651 (2018; Zbl 1394.81020); Soft Comput. 22, No. 8, 2485–2493 (2018; Zbl 1398.06012)], is how it affects spectra of observables, particularly, their extremal points. We describe boundaries for extremal points of the spectrum of the sum of observables in a general case, and we give necessary and sufficient conditions under which the spectrum attains these boundary values. Moreover, we show that every bounded observable \(x\) on a complete MV-effect algebra \(E\) can be decomposed into the sum \(x=\tilde{x}+x'\), where \(\tilde{x}\) is the greatest sharp observable less than \(x\) and \(x'\) is a meager and extremally non-invertible observable.

MSC:
06D35 MV-algebras
03G12 Quantum logic
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