Skip to main content
SpringerLink
Log in

Generalized Bosbach states: Part II

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

We continue the investigation of generalized Bosbach states that we began in Part I, restricting our research to the commutative case and treating further aspects related to these states. Part II is concerned with similarity convergences, continuity of states and the construction of the s-completion of a commutative residuated lattice, where s is a generalized Bosbach state.

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. Balbes R., Dwinger P.: Distributive Lattices. University of Missouri Press, Columbia (1974)

    MATH  Google Scholar 

  2. Bělohlávek R.: Fuzzy Relational Systems. Foundations and Principles. Kluwer Academic/Plenum Publishers, New York/Boston/Dordrecht/London/Moscow (2002)

    Book  MATH  Google Scholar 

  3. Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. In: Trends in Logic-Studia Logica Library 7, Kluwer Academic Publishers, Dordrecht (2000)

  4. Ciungu, L.C.: Algebraic Models for Multiple-valued Logics. Ph.D. thesis, Bucharest University, Bucharest (2007)

  5. Ciungu L.C.: Bosbach and Riečan states on residuated lattices. J. Appl. Funct. Anal. 2, 175–188 (2008)

    MathSciNet  Google Scholar 

  6. Ciungu, L.C., Georgescu, G., Mureşan, C.: Generalized Bosbach states: Part I. Arch. Math. Log 52(3-4), 335–376 (2013) doi:10.1007/s00153-012-0319-2

    Google Scholar 

  7. Dvurečenskij A., Rachůnek J.: Probabilistic averaging in bounded Rl-monoids. Semigroup Forum 72, 191–206 (2006)

    Article  MATH  Google Scholar 

  8. Dvurečenskij A., Rachůnek J.: On Riečan and Bosbach states for bounded non-commutative Rl-monoids. Math. Slovaca 56(5), 487–500 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Flaminio T., Godo L.: A logic for reasoning about the probability of fuzzy events. Fuzzy Sets Syst. 158(6), 625–638 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Flaminio T.: Strong non-standard completeness for fuzzy logic. Soft Comput. 12, 321–333 (2008)

    Article  MATH  Google Scholar 

  11. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. In: Studies in Logic and The Foundations of Mathematics 151, Elsevier, Amsterdam/Boston/Heidelberg/London/NewYork/Oxford/Paris/San Diego/San Francisco/Singapore/ Sydney/Tokyo (2007)

  12. Georgescu G.: Bosbach states on fuzzy structures. Soft Comput. 8, 217–230 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Georgescu G., Popescu A.: Similarity convergence in residuated structures. Log. J. IGPL 13(4), 389–413 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Georgescu, G., Mureşan, C.: Generalized Bosbach States (2010), available online at http://arxiv.org/abs/1007.2575

  15. Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. American Mathematical Society, Providence, RI (1986)

  16. Hájek P.: Metamathematics of Fuzzy Logic, Trends in Logic-Studia Logica. Kluwer Academic Publishers, Dordrecht/Boston/London (1998)

    Book  Google Scholar 

  17. Iorgulescu A.: Algebras of Logic as BCK Algebras. Editura ASE, Bucharest (2008)

    MATH  Google Scholar 

  18. Klement E.P., Mesiar R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  19. Kowalski, T., Ono, H.: Residuated lattices: an algebraic glimpse at logics without contraction. manuscript (2000)

  20. Leuştean, I.: Contributions to the Theory of MV-algebras. Ed. Universitară, Bucharest (2009)

  21. Leuştean I.: Metric Completions of MV-algebras with States: An Approach to Stochastic Independence. J. Log. Comput. Adv. Access. Published online on March 10 (2009)

  22. Mureşan, C.: Characterization of the reticulation of a residuated lattice. J. Multiple-valued Log. Soft Comput. 16, No. 3–5 (2010), Special Issue: Multiple-Valued Logic and Its Algebras, pp. 427–447

  23. Piciu D.: Algebras of Fuzzy Logic. Editura Universitaria Craiova, Craiova (2007)

    Google Scholar 

  24. Riečan, B., Mundici, D.: Probability in MV–algebras. In: Pap, E. (eds). Handbook of Measure Theory, North– Holland E. Pap Ed, pp. 869—909 (2002)

  25. Turunen, E.: Mathematics behind Fuzzy Logic. Advances in Soft Computing, Physica–Verlag, Heidelberg (1999)

  26. Wang G.J., Leung Y.: Integrated semantics and logic metric spaces. Fuzzy Sets Syst. 136, 71–91 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang G.J., Hui X.J., Song J.S.: The R0-type fuzzy logic metric space and an algorithm for solving fuzzy modus ponens. Comput. Math. Appl. 55(9), 1977–1987 (2008)

    MathSciNet  Google Scholar 

  28. Zhang, J.: Metric Structures on some MTL-algebras and its applications. In: 2010 Seventh International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2010). pp. 231–237

  29. Zhou H., Zhao B.: Generalized Bosbach and Riečan states based on relative negations in residuated lattices. Fuzzy Sets Syst. 187(1), 33–57 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA, 52242-1419, USA

    Lavinia Corina Ciungu

  2. Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 010014, Bucharest, Romania

    George Georgescu & Claudia Mureşan

Authors
  1. Lavinia Corina Ciungu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. George Georgescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Claudia Mureşan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudia Mureşan.

Additional information

Dedicated to the memory of Professor Nicolae Popescu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ciungu, L.C., Georgescu, G. & Mureşan, C. Generalized Bosbach states: Part II. Arch. Math. Logic 52, 707–732 (2013). https://doi.org/10.1007/s00153-013-0339-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-013-0339-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation