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Discrete (n + 1)-valued states and n-perfect pseudo-effect algebras

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Abstract

We give sufficient and necessary conditions to guarantee that a pseudo-effect algebra admits an (n + 1)-valued discrete state. We introduce n-perfect pseudo-effect algebras as algebras which can be split into n + 1 comparable slices. We prove that the category of strong n-perfect pseudo-effect algebras is categorically equivalent to the category of torsion-free directed partially ordered groups of a special type.

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References

  • Adámek J, Herrlich H, Strecker GE (1990) Abstract and concrete categories: the joy of facts. Wiley, New York (republished in: Reprints in theory and applications of categories, no. 17, pp 1–507. http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf

  • Belluce LP, Di Nola A, Letieri A (1993) Local MV-algebras. Rend Circ Matem Palermo 42:347–361

    Article  MATH  Google Scholar 

  • Bennett MK, Foulis DJ (1997) Interval and scale effect algebras. Adv Appl Math 19:200–215

    Article  MathSciNet  MATH  Google Scholar 

  • Buhagiar D, Chetcuti E, Dvurečenskij A (2006) Loomis-Sikorski representation of monotone σ-complete effect algebras. Fuzzy Sets and Systems 157:683–690

    Article  MATH  Google Scholar 

  • Di Nola A, Lettieri A (1994) Perfect MV-algebras are categorical equivalent to Abelian ℓ-groups. Stud Logica 53:417–432

    Article  MathSciNet  MATH  Google Scholar 

  • Di Nola A, Dvurečenskij A, Hyčko M., Manara C (2005) Entropy on effect algebras with the Riesz decomposition property II: MV-algebras. Kybernetika 41:161–176

    MathSciNet  MATH  Google Scholar 

  • Di Nola A, Dvurečenskij A, Tsinakis C (2008) On perfect GMV-algebras. Commun Algebra 36:1221–1249

    Article  MATH  Google Scholar 

  • Dvurečenskij A (1993) Gleason’s theorem and its applications. Kluwer Academic Publisher, Dordrecht

  • Dvurečenskij A (2002) Pseudo MV-algebras are intervals in ℓ-group. J Aust Math Soc 72:427–445

    Article  MathSciNet  MATH  Google Scholar 

  • Dvurečenskij A (2003) Ideals of pseudo-effect algebras and their applications. Tatra Mt Math Publ 27:45–65

    MATH  Google Scholar 

  • Dvurečenskij A (2004) States and radicals of pseudo-effect algebras. Atti Semin Matem Fis Univ Modena 52:85–103

    Google Scholar 

  • Dvurečenskij A (2007) Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups. Journal of the Australian Mathematical Society 82:183–207

    Article  MathSciNet  MATH  Google Scholar 

  • Dvurečenskij A (2008) On n-perfect GMV-algebras. J Algebra 319:4921–4946

    Article  MathSciNet  MATH  Google Scholar 

  • Dvurečenskij A (2010) Cyclic elements and subalgebras of GMV-algebras. Soft Comput 14:257–264

    Article  MATH  Google Scholar 

  • Dvurečenskij A, Pulmannová S (2000) “New Trends in Quantum Structures”. Kluwer Academic Publishers/Ister Science, Dordrecht/Bratislava

    Book  MATH  Google Scholar 

  • Dvurečenskij A, Vetterlein T (2001a) Pseudoeffect algebras. I. Basic properties. Int J Theor Phys 40:685–701

    Article  MATH  Google Scholar 

  • Dvurečenskij A, Vetterlein T (2001b) Pseudoeffect algebras. II. Group representation. Int J Theor Phys 40:703–726

    Article  MATH  Google Scholar 

  • Dvurečenskij A, Vetterlein T (2002) Algebras in the positive cone of po-groups. Order 19:127–146

    Article  MathSciNet  MATH  Google Scholar 

  • Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1325–1346

    Article  MathSciNet  Google Scholar 

  • Fuchs L (1963) Partially Ordered Algebraic Systems. Pergamon Press, Oxford-New York

    MATH  Google Scholar 

  • Georgescu G, Iorgulescu A (2001) Pseudo-MV algebras. Multiple Valued Log Int J 6:95–135

    Google Scholar 

  • Glass AMW (1999) “Partially Ordered Groups”. World Scientific, Singapore, New-Jersey, London, Hong Kong

    Book  MATH  Google Scholar 

  • Goodearl KR (1986) Partially ordered abelian groups with interpolation. In: Mathematical surveys and monographs no 20. American Mathematical Society, Providence

  • Hájek P (2003) Observations on non-commutative fuzzy logic. Soft Comput 8:38–43

    Article  MATH  Google Scholar 

  • Kalmbach G (1983) “Orthomodular Lattices”. Academic Press, London

    MATH  Google Scholar 

  • Kôpka F, Chovanec F (1994) D-posets. Math Slovaca 44:21–34

    MathSciNet  MATH  Google Scholar 

  • Li Y (2008) Structures of scale generalized effect algebras and scale effect algebras. Acta Math Sin 51:863–876 (In Chinese)

    MATH  Google Scholar 

  • Rachunek J (2002) A non-commutative generalization of MV-algebras. Czechoslov Math J 52:255–273

    Article  MathSciNet  MATH  Google Scholar 

  • Riečanová Z (2008) Effect algebraic extensions of generalized effect algebras and two-valued states. Fuzzy Sets Syst 159:1116–1122

    Article  MATH  Google Scholar 

  • Riečanová Z, Marinová I (2005) Generalized homogeneous, prelattice and MV-effect algebras. Kybernetika 41:129–142

    MathSciNet  MATH  Google Scholar 

  • Xie Y, Li Y (2010) Riesz ideals in generalized pseudo-effect algebra and their unitizations. Soft Comput 14:387–398

    Article  MATH  Google Scholar 

  • Xie Y, Li Y, Guo J, Ren F, Li D (2011) Weak commutative pseudo-effect algebras. Int J Theor Phys 50:1186–1197

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors thank for the support by SAIA, n.o. (Slovak Academic Information Agency) and the Ministry of Education, Science, Research and Sport of the Slovak Republic. This work is also supported by National Science Foundation of China (Grant No. 11201279), and the Fundamental Research Funds for the Central Universities (Grant No. GK201002037). A.D. thanks for the support by Center of Excellence SAS - Quantum Technologies -, ERDF OP R&D Project meta-QUTE ITMS 26240120022, Slovak Research and Development Agency under the contract APVV-0178-11, the grant VEGA No. 2/0059/12 SAV and by CZ.1.07/2.3.00/20.0051. The authors are very indebted to an anonymous referee for his/her careful reading and suggestions which helped us to improve the readability of the paper.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij & Yongjian Xie

  2. Department of Algebra and Geometry, Palacký University, 771 46, Olomouc, Czech Republic

    Anatolij Dvurečenskij

  3. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, China

    Yongjian Xie

  4. College of Science, Xi’an University of Science and Technology, Xi’an, 710054, China

    Aili Yang

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  1. Anatolij Dvurečenskij
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  2. Yongjian Xie
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  3. Aili Yang
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Correspondence to Anatolij Dvurečenskij.

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Communicated by L. Spada.

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Dvurečenskij, A., Xie, Y. & Yang, A. Discrete (n + 1)-valued states and n-perfect pseudo-effect algebras. Soft Comput 17, 1537–1552 (2013). https://doi.org/10.1007/s00500-013-1001-2

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