Soft Computing

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

Sum of observables on MV-effect algebras

  • Anatolij Dvurečenskij


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.


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



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.


  1. Bauer H (1996) Probability theory. de Gruyter, BerlinCrossRefzbMATHGoogle Scholar
  2. Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–834MathSciNetCrossRefzbMATHGoogle Scholar
  3. Butnariu D, Klement EP (1993) Triangular norm-based measures and games with fuzzy coalitions. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  4. Cignoli R, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  5. Dvurečenskij A (2003) Central elements and Cantor–Bernstein’s theorem for pseudo-effect algebras. J Aust Math Soc 74:121–143MathSciNetCrossRefzbMATHGoogle Scholar
  6. Dvurečenskij A (2014) Representable effect algebras and observables. Int J Theor Phys 53:2855–2866. doi: 10.1007/s10773-014-2083-z MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dvurečenskij A (2016a) Olson order of quantum observables. Int J Theor Phys 55:4896–4912. doi: 10.1007/s10773-016-3113 CrossRefzbMATHGoogle Scholar
  8. Dvurečenskij A (2016b) Quantum observables and effect algebras.
  9. Dvurečenskij A, Kuková M (2014) Observables on quantum structures. Inf Sci 262:215–222. doi: 10.1016/j.ins.2013.09.014 MathSciNetCrossRefzbMATHGoogle Scholar
  10. Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer, Dordrecht, Ister Science, Bratislava, 541 + xvi ppGoogle Scholar
  11. Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1325–1346MathSciNetCrossRefzbMATHGoogle Scholar
  12. Fuchs L (1963) Partially ordered algebraic systems. Pergamon Press, OxfordzbMATHGoogle Scholar
  13. Goodearl KR (1986) Partially ordered abelian groups with interpolation. Mathematical Surveys and Monographs No. 20, American Mathematical Society, ProvidenceGoogle Scholar
  14. Gudder SP (1966) Uniqueness and existence properties of bounded observables. Pac J Math 19:81–93MathSciNetCrossRefzbMATHGoogle Scholar
  15. Halmos PR (1974) Measure theory. Springer, BerlinzbMATHGoogle Scholar
  16. Jenča G (2010) Sharp and meager elements in orthocomlete homogeneous effect algebras. Order 27:41–61MathSciNetCrossRefzbMATHGoogle Scholar
  17. Jenča G, Riečanová Z (1999) On sharp elements in lattice ordered effect algebras. Busefal 80:24–29Google Scholar
  18. Jenčová A, Pulmannová S, Vinceková E (2011) Observables on \(\sigma \)-MV-algebras and \(\sigma \)-lattice effect algebras. Kybernetika 47:541–559MathSciNetzbMATHGoogle Scholar
  19. Kadison R (1951) Order properties of bounded self-adjoint operators. Proc Am Math Soc 2:505–510MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kallenberg O (1997) Foundations of modern probability. Springer, New YorkzbMATHGoogle Scholar
  21. Olson MP (1971) The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice. Proc Am Math Soc 28:537–544CrossRefzbMATHGoogle Scholar
  22. Pulmannová S (2002) Compatibility and decomposition of effects. J Math Phys 43:2817–2830MathSciNetCrossRefzbMATHGoogle Scholar
  23. Ravindran K (1996) On a structure theory of effect algebras, Ph.D. thesis. Kansas State University, ManhattanGoogle Scholar
  24. Riečanová Z (2000) Generalization of blocks for d-lattice and lattice ordered effect algebras. Int J Theor Phys 39:231–237MathSciNetCrossRefzbMATHGoogle Scholar
  25. Sikorski R (1964) Boolean algebras. Springer, BerlinzbMATHGoogle Scholar
  26. Varadarajan VS (1968) Geometry of quantum theory, vol 1. van Nostrand, PrincetonCrossRefzbMATHGoogle Scholar

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© 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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