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Soft Computing

, Volume 22, Issue 24, pp 8041–8049 | Cite as

The spectrum of the sum of observables on \(\sigma \)-complete MV-effect algebras

  • Jiří Janda
  • Yongjian Xie
Foundations
  • 53 Downloads

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.

Keywords

Monotone \(\sigma \)-complete effect algebra Observable on a \(\sigma \)-complete MV-effect algebra Sum of observables Spectrum of an observable Sharp observable Meager observable 

Notes

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).

Compliance with ethical standards

Conflict of interest

Author Jiří Janda declares that he has no conflict of interest. Author Yongjian Xie declares that he has no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceMasaryk UniversityBrnoCzech Republic
  2. 2.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China

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