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Signed \(\lambda \)-Measures on Effect Algebras

  • Akhilesh Kumar SinghEmail author
Research Article

Abstract

The concepts of \(\lambda \)-measures and signed \(\lambda \)-measures (nonadditive measures) defined on effect algebras are introduced and investigated in this paper. A few examples of \(\lambda \)-measures and signed \(\lambda \)-measures are also given. Moreover, the Jordan decomposition theorem for signed \(\lambda \)-measures defined on effect algebras is also presented.

Keywords

\(\lambda \)-measures Signed \(\lambda \)-measure Jordan decomposition theorem Effect algebras 

Mathematics Subject Classification

06A11 28A12 28E99 06C15. 

References

  1. 1.
    Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall, Elsevier, Upper Saddle RiverzbMATHGoogle Scholar
  2. 2.
    Pap E (2002) Handbook of measure theory. Elsevier, North-Holland, AmsterdamzbMATHGoogle Scholar
  3. 3.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefGoogle Scholar
  4. 4.
    Sugeno M (1974) Theory of Fuzzy Integrals and Its Applications. Ph.D, Dis- sertation, Tokyo Institute of TechnologyGoogle Scholar
  5. 5.
    Riečanova Z, Zajac M, Pulmanova S (2011) Effect algebras of positive linear operators densly defined on Hilbert space. Rep Math Phys 68:261–270ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Nguyen HT, Kreinovich V, Lorkowski J, Abu S (2015) Why Sugeno lambda- measures. Departmental Technical Reports (CS), Paper 906Google Scholar
  7. 7.
    Bennet MK, Foulis DJ (1994) Effect algebras and unsharp quantum logics. Found Phys 24(10):1331–1352ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Kôpka F, Chovaneć F (1992) D-posets of fuzzy sets. Tetra Mount Math Publ 1:83–87MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dvurečenskij A, Pulmannová S (2000) New Trends in Quantum Structures. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
  10. 10.
    Beltrametti EG, Cassinelli G (1981) The logic of quantum mechanics. Addison-Wesley Publishing Company, ReadingzbMATHGoogle Scholar
  11. 11.
    Dvurečenskij A (2011) The lattice and simplex structure of states on pseudo effect algebras. Int J Theor Phys 50:2758–2775MathSciNetCrossRefGoogle Scholar
  12. 12.
    Avallone A, Basile A (2003) On a Marinacci uniqueness theorem for mea- sures. J Math Anal Appl 286:378–390MathSciNetCrossRefGoogle Scholar
  13. 13.
    Avallone A, Simone A De, Vitolo P (2006) Effect algebras and extensions of measures. Bullenttino U M I 9-B(8):423-444Google Scholar
  14. 14.
    Bennet MK, Foulis DJ (1995) Phi-symmetric effect algebras. Found Phys 25:1699–1722ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Bennet MK, Foulis DJ, Greechie RJ (1994) Sums and products of interval algebras. Int J Theor Phys 33:2114–2136MathSciNetGoogle Scholar
  16. 16.
    Guintini R, Greuling R (1989) Towards a formal language for unsharp prop- erties. Found Phys 19:931–945ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Khare M, Singh AK (2008) Atom and a Saks type decomposition in effect algebras. Demonstr Math 38(1):59–70ADSMathSciNetzbMATHGoogle Scholar
  18. 18.
    Khare M, Singh AK (2008) Atoms and Dobrakov submeasures in effect algebras. Fuzzy Sets Syst 159:1123–1128MathSciNetCrossRefGoogle Scholar
  19. 19.
    Khare M, Singh AK (2008) Weakly tight functions, their Jordan type decom- position and total variation in effect algebras. J Math Anal Appl 344(1):535–545MathSciNetCrossRefGoogle Scholar
  20. 20.
    Khare M, Singh AK (2008) Pseudo-atoms, atoms and a Jordan type decom- position in effect algebras. J Math Anal Appl 344(1):238–252MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chitesu I (2015) Why \(\lambda \)-additive (fuzzy) measures? Kybernetika 51(2):246–254MathSciNetGoogle Scholar
  22. 22.
    Qiang Z, A class of nonadditive signed measures. preprint: 1-13Google Scholar
  23. 23.
    Wang Z, Klir G (2009) Generalized Measure Theory. Springer, BerlinCrossRefGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2018

Authors and Affiliations

  1. 1.Kamla Nehru Institute of TechnologySultanpurIndia

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