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Lexicographic effect algebras

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Abstract

We investigate a class of effect algebras that can be represented in the form \({\Gamma (H \overrightarrow{\times} G}\), (u, 0)), where \({H \overrightarrow{\times} G}\) means the lexicographic product of an Abelian unital po-group (H, u) and an Abelian directed po-group G. We study conditions when an effect algebra is of this form. Fixing a unital po-group (H, u), the category of strongly (H, u)-perfect effect algebras is introduced and it is shown that it is categorically equivalent to the category of directed po-groups with interpolation. We prove some representation theorems of lexicographic effect algebras, including a subdirect product representation by antilattice lexicographic effect algebras.

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References

  1. Birkhoff G.: Lattice Theory. Amer. Math. Soc. Coll. Publ., vol. 25. American Mathematical Society, Providence (1967)

    Google Scholar 

  2. Cignoli R., Torrens A.: Retractive MV-algebras. Mathware Soft Comput. 2, 157–165 (1995)

    MathSciNet  MATH  Google Scholar 

  3. Diaconescu D., Flaminio T., Leuştean I.: Lexicographic MV-algebras and lexicographic states. Fuzzy Sets and Systems 244, 63–85 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Di Nola A., Lettieri A.: Perfect MV-algebras are categorically equivalent to abelian l-groups. Studia Logica 53, 417–432 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Di Nola A., Lettieri A.: Coproduct MV-algebras, nonstandard reals and Riesz spaces. J. Algebra 185, 605–620 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Dvurečenskij A.: Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups. J. Aust. Math. Soc. 82, 183–207 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Dvurečenskij A.: \({\mathbb{H}}\)-perfect pseudo MV-algebras and their representations. Math. Slovaca 65, 761–788 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Dvurečenskij, A.: Pseudo MV-algebras and lexicographic product. Fuzzy Sets and Systems (to appear) DOI:10.1016/j.fss.2015.09.024

  11. Dvurečenskij A., Kolařík M.: Lexicographic product vs \({\mathbb{Q}}\)-perfect and \({\mathbb{H}}\)-perfect pseudo effect algebras. Soft Computing 17, 1041–1053 (2014)

    Article  MATH  Google Scholar 

  12. Dvurečenskij , A. , Kolařík J.: The lexicographic product of po-groups and n-perfect pseudo effect algebras. Internat. J. Theoret. Phys. 52, 2760–2772 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dvurečenskij A., Pulmannová S.: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht and Ister Science, Bratislava (2000)

    Book  MATH  Google Scholar 

  14. Dvurečenskij A., Vetterlein T.: Pseudoeffect algebras. I. Basic properties. Internat. J. Theoret. Phys. 40, 685–701 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  16. Dvurečenskij A., Vetterlein T.: Congruences and states on pseudo-effect algebras. Found. Phys. Letters 14, 425–446 (2001)

    Article  MATH  Google Scholar 

  17. Dvurečenskij A., Vetterlein T.: Archimedeanness and the MacNeille completion of pseudoeffect algebras and po-groups. Algebra Universalis 50, 207–230 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dvurečenskij A., Xie Y., Yang A.: Discrete (n + 1)-valued states and n-perfect pseudo-effect algebras. Soft Computing 17, 1537–1552 (2013)

    Article  MATH  Google Scholar 

  19. Foulis D.J., Bennett M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331–1352 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fuchs L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford (1963)

    MATH  Google Scholar 

  21. Georgescu G., Iorgulescu A.: Pseudo-MV algebras. Multiple Val. Logic 6, 95–135 (2001)

    MathSciNet  MATH  Google Scholar 

  22. Glass A.M.W.: Partially Ordered Groups. World Scientific, Singapore (1999)

    Book  MATH  Google Scholar 

  23. Goodearl K.R.: Partially Ordered Abelian Groups with Interpolation. Math. Surveys and Monographs No. 20. . American Mathematical Society, Providence (1986)

    Google Scholar 

  24. Luxemburg W.A.J., Zaanen A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)

    MATH  Google Scholar 

  25. Mac Lane S.: Categories for the Working Mathematician. Springer, New York (1971)

    Book  MATH  Google Scholar 

  26. Mundici D.: Interpretation of AF C*-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pulmannová S.: Compatibility and decomposition of effects. J. Math. Phys. 43, 2817–2830 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ravindran, K.: On a Structure Theory of Effect Algebras. PhD thesis, Kansas State University (1996)

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Authors and Affiliations

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

    Anatolij Dvurečenskij

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  1. Anatolij Dvurečenskij
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Correspondence to Anatolij Dvurečenskij.

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Presented by S. Pulmannova.

This work was supported by the Slovak Research and Development Agency under contract APVV-0178-11, grant VEGA No. 2/0059/12 SAV, CZ.1.07/2.3.00/20.0051, and GAČR 15-15286S.

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Dvurečenskij, A. Lexicographic effect algebras. Algebra Univers. 75, 451–480 (2016). https://doi.org/10.1007/s00012-016-0374-3

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