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Direct summands and retract mappings of generalized MV-algebras

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Abstract

In the present paper we deal with generalized MV-algebras (GMV-algebras, in short) in the sense of Galatos and Tsinakis. According to a result of the mentioned authors, GMV-algebras can be obtained by a truncation construction from lattice ordered groups. We investigate direct summands and retract mappings of GMV-algebras. The relations between GMV-algebras and lattice ordered groups are essential for this investigation.

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References

  1. C. C. Chang: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88 (1958), 467–490.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Cignoli, I. M. I. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publishers, Dordrecht, 2000.

    MATH  Google Scholar 

  3. A. Dvurečenskij: Pseudo MV-algebras are intervals in ℓ-groups. J. Austral. Math. Soc. 72 (2004), 427–445.

    Article  Google Scholar 

  4. L. Fuchs:: Infinite Abelian Groups, Vol. 1. Academic Press, New York and London, 1970.

    MATH  Google Scholar 

  5. N. Galatos and C. Tsinakis: Generalized MV-algebras. J. Algebra 283 (2005), 254–291.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Georgescu and A. Iorgulescu: Pseudo MV-algebras: a noncommutative extension of MV-algebras. In: Information Technology, Bucharest 1999, INFOREC, Bucharest, 1999, pp. 961–968.

    Google Scholar 

  7. G. Georgescu and A. Iorgulescu: Pseudo MV-algebras. Multi-Valued Logic 6 (2001), 95–135.

    MathSciNet  MATH  Google Scholar 

  8. A. M. W. Glass: Partially Ordered Groups. World Scientific, Singapore-New Jersey-London-Hong Kong, 1999.

    MATH  Google Scholar 

  9. J. Jakubík: Direct product decompositions of MV-algebras. Czech. Math. J. 44 (1994), 725–739.

    MATH  Google Scholar 

  10. J. Jakubík: Direct product decompositions of pseudo MV-algebras. Archivum Math. 37 (2001), 131–142.

    MATH  Google Scholar 

  11. J. Jakubík: Weak \( (\mathfrak{m},\mathfrak{n}) \)-distributivity of lattice ordered groups and of generalized MV-algebras. Soft Computing 10 (2006), 119–124.

    Article  MATH  Google Scholar 

  12. J. Jakubík: On interval subalgebras of generalized MV-algebras. Math. Slovaca 56 (2006), 387–395.

    MathSciNet  MATH  Google Scholar 

  13. J. Jakubík: Retracts of abelian lattice ordered groups. Czech. Math. J. 39 (1989), 477–489.

    Google Scholar 

  14. J. Jakubík: Retract varieties of lattice ordered groups. Czech. Math. J. 40 (1990), 104–112.

    Google Scholar 

  15. J. Jakubík: Complete retract mappings of a complete lattice ordered group. Czech. Math. J. 43 (1993), 309–318.

    MATH  Google Scholar 

  16. J. Jakubík: On absolute retracts and absolute convex retracts in some classes of ℓ-groups. Discussiones Math., Gener. Alg. Appl. 23 (2003), 19–30.

    MATH  Google Scholar 

  17. J. Jakubík: Retract mappings of projectable MV-algebras. Soft Computing 4 (2000), 27–32.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Rachůnek: A non-commutative generalization of MV-algebras. Czech. Math. J. 52 (2002), 255–273.

    Article  MATH  Google Scholar 

  20. J. Rachůnek and D. Šalounová: Direct product factors in GMV-algebras. Math. Slovaca 55 (2005), 399–407.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, SK-040 01, Košice, Slovakia

    Ján Jakubík

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  1. Ján Jakubík
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Correspondence to Ján Jakubík.

Additional information

Supported by VEGA Agency grant 1/2002/05.

This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information, grant I/2/2005.

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Jakubík, J. Direct summands and retract mappings of generalized MV-algebras. Czech Math J 58, 183–202 (2008). https://doi.org/10.1007/s10587-008-0013-z

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  • DOI: https://doi.org/10.1007/s10587-008-0013-z

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