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Soft Computing

, Volume 22, Issue 8, pp 2485–2493 | Cite as

Sum of observables on MV-effect algebras

  • Anatolij Dvurečenskij
Foundations

Abstract

Using a one-to-one correspondence between observables and their spectral resolutions, we introduce the sum of any two bounded observables of a \(\sigma \)-MV-effect algebra. This sum is commutative, associative with neutral element. Under the Olson order of observables, the set of bounded observables is a partially ordered semigroup, and the set of sharp observables is even a Dedekind \(\sigma \)-complete \(\ell \)-group with strong unit.

Keywords

Effect algebra MV-effect algebra Monotone \(\sigma \)-complete effect algebra Observable Sharp observable Spectral resolution Sum of observables Olson order \(\ell \)-Group Semigroup 

Notes

Acknowledgements

The author is very indebted to anonymous referees for their careful reading and suggestions which helped us to improve the readability of the paper. This study was funded by the Grants VEGA Nos. 2/0069/16 SAV and GAČR 15-15286S.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia
  2. 2.Department of Algebra GeometryPalacký UniversityOlomoucCzechia

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