Abstract
The natural question about the sum of observables on \(\sigma \)-complete MV-effect algebras, which was recently defined by A. Dvurečenskij, 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.
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Funding
This study was funded by the Czech Science Foundation, project Algebraic, many-valued and quantum structures for uncertainty modeling (Grant Number GA15-15286S) and by the National Science Foundation of China (Grant Number 61673250).
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Author Jiří Janda declares that he has no conflict of interest. Author Yongjian Xie declares that he has no conflict of interest.
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This article does not contain any studies with human participants or animals performed by any of the authors.
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Communicated by A. Di Nola.
The authors acknowledge the support (J.J.) by the CSF, project Algebraic, many-valued and quantum structures for uncertainty modeling, GA15-15286S and (Y.X.) by the National Science Foundation of China (Grant No. 61673250). We also thank to anonymous referees for their careful reading and valuable comments which improved the quality of the article.
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Janda, J., Xie, Y. The spectrum of the sum of observables on \(\sigma \)-complete MV-effect algebras. Soft Comput 22, 8041–8049 (2018). https://doi.org/10.1007/s00500-018-3078-0
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DOI: https://doi.org/10.1007/s00500-018-3078-0