Skip to main content
SpringerLink
Log in

Generalizations of pseudo MV-algebras and generalized pseudo effect algebras

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

We deal with unbounded dually residuated lattices that generalize pseudo MV-algebras in such a way that every principal order-ideal is a pseudo MV-algebra. We describe the connections of these generalized pseudo MV-algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo MV-algebra A by means of the positive cone of a suitable ℓ-group G A . We prove that the lattice of all (normal) ideals of A and the lattice of all (normal) convex ℓ-subgroups of G A are isomorphic. We also introduce the concept of Archimedeanness and show that every Archimedean generalized pseudo MV-algebra is commutative.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Anderson and T. Feil: Lattice-Ordered Groups (An Introduction). D. Reidel, Dordrecht, 1988.

    MATH  Google Scholar 

  2. P. Bahls, J. Cole, N. Galatos, P. Jipsen and C. Tsinakis: Cancellative residuated lattices. Algebra Univers. 50 (2003), 83–106.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Bigard, K. Keimel and S. Wolfenstein: Groupes et Anneaux Réticulés. Springer, Berlin, 1977.

    MATH  Google Scholar 

  4. R. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer Acad. Publ., Dordrecht, 2000.

    MATH  Google Scholar 

  5. A. Dvurečenskij: Pseudo MV-algebras are intervals in ℓ-groups. J. Austral. Math. Soc. (Ser. A) 72 (2002), 427–445.

    MATH  Google Scholar 

  6. A. Dvurečenskij and S. Pulmannová: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht, 2000.

    MATH  Google Scholar 

  7. A. Dvurečenskij and J. Rachğnek: Probabilistic averaging in bounded Rℓ-monoids. Semi-group Forum 72 (2006), 191–206.

    Article  MATH  Google Scholar 

  8. A. Dvurečenskij and T. Vetterlein: Pseudo-effect algebras I. Basic properties. Internat. J. Theor. Phys. 40 (2001), 685–701.

    Article  MATH  Google Scholar 

  9. A. Dvurečenskij and T. Vetterlein: Pseudo-effect algebras II. Group representations. Internat. J. Theor. Phys. 40 (2001), 703–726.

    Article  MATH  Google Scholar 

  10. A. Dvurečenskij and T. Vetterlein: Generalized pseudo-effect algebras. In: Lectures on Soft Computing and Fuzzy Logic (A. Di Nola, G. Gerla, eds.), Springer, Berlin, 2001, pp. 89–111.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Georgescu and A. Iorgulescu: Pseudo-MV algebras. Mult.-Valued Log. 6 (2001), 95–135.

    MathSciNet  MATH  Google Scholar 

  13. G. Georgescu, L. Leuştean and V. Preoteasa: Pseudo-hoops. J. Mult.-Val. Log. Soft Comput. 11 (2005), 153–184.

    MATH  Google Scholar 

  14. A. M. W. Glass: Partially Ordered Groups. World Scientific, Singapore, 1999.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. A. Iorgulescu: Classes of pseudo-BCK(pP) lattices. Preprint.

  17. P. Jipsen and C. Tsinakis: A survey of residuated lattices. In: Ordered Algebraic Structures (J. Martines, ed.), Kluwer Acad. Publ., Dordrecht, 2002, pp. 19–56.

    Google Scholar 

  18. J. Kühr: Ideals of noncommutative DRℓ-monoids. Czech. Math. J. 55 (2005), 97–111.

    Article  MATH  Google Scholar 

  19. J. Kühr: Finite-valued dually residuated lattice-ordered monoids. Math. Slovaca 56 (2006), 397–408.

    MathSciNet  MATH  Google Scholar 

  20. J. Kühr: On a generalization of pseudo MV-algebras. J. Mult.-Val. Log. Soft Comput 12 (2006), 373–389.

    MATH  Google Scholar 

  21. T. Kovář: General Theory of Dually Residuated Lattice Ordered Monoids. Ph.D. thesis, Palacký Univ., Olomouc, 1996.

    Google Scholar 

  22. J. Martinez: Archimedean lattices.. Algebra Univers. 3 (1973), 247–260.

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  25. J. Rachůnek: Prime spectra of non-commutative generalizations of MV-algebras. Algebra Univers. 48 (2002), 151–169.

    Article  MATH  Google Scholar 

  26. K. L. N. Swamy: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Algebra and Geometry, Faculty of Science, Palacký University, Tomkova 40, 779 00, Olomouc, Czech Republic

    Jan Kühr

Authors
  1. Jan Kühr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Kühr.

Additional information

Supported by the Research and Development Council of the Czech Govenrment via the project MSM6198959214.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kühr, J. Generalizations of pseudo MV-algebras and generalized pseudo effect algebras. Czech Math J 58, 395–415 (2008). https://doi.org/10.1007/s10587-008-0023-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-008-0023-x

Keywords

Search

Navigation