Skip to main content
SpringerLink
Log in

Natural negation of interval-valued t-(co) norms and implications

  • Published:
Fuzzy Optimization and Decision Making Aims and scope Submit manuscript

Abstract

In this paper, we investigate interval-valued fuzzy negations induced by interval-valued \(t\)-norms, \(t\)-conorms or implications. Some properties of interval-valued fuzzy negations induced by interval-valued sup-morphism \(t\)-norms, inf-morphism \(t\)-conorms or \(R\)-implications are firstly obtained. We also show interval-valued automorphisms acting on the interval-valued fuzzy negations induced by interval-valued \(t\)-norms, \(t\)-conorms or implications. Finally, the relations among the interval-valued fuzzy negations induced by interval-valued \(t\)-norms, \(t\)-conorms or implications are explored.

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

Fig. 1

Similar content being viewed by others

References

  • Acióly, B. M., & Bedregal, B. C. (1997). A quasi-metric topology compatible with inclusion-monotonicity property on interval space. Reliable Computing, 3(3), 305–313.

    Article  MathSciNet  MATH  Google Scholar 

  • Baczyński, M., & Jayaram, B. (2008). (S, N)- and R-implications: A state-of-the-art survey. Fuzzy Sets and Systems, 159, 1836–1859.

    Article  MathSciNet  MATH  Google Scholar 

  • Bedregal, B. C. (2010). On interval fuzzy negations. Fuzzy Sets and Systems, 161(17), 2290–2313.

    Article  MathSciNet  MATH  Google Scholar 

  • Bedregal, B. R. C., Takahashi, A. (2005). Interval t-norms as interval representations of t-norms, The 14th IEEE International Conference on Fuzzy Systems, May, 25–25, pp. 909–914.

  • Birkhof, G. (1948). Lattice theory, American Mathematical Society Colloquium Publications, vol. 25, revised edition. New York: American Mathematical Society.

  • Bustince, H., Burillo, P., & Soria, F. (2003). Automorphism, negations and implication operators. Fuzzy Sets and Systems, 134, 209–229.

    Article  MathSciNet  MATH  Google Scholar 

  • Cornelis, C., Deschrijver, G., & Kerre, E. E. (2004). Implication in intuitionistic and interval-valued fuzzy set theory: Construction, classification and application. International Journal of Approximate Reasoning, 35, 55–95.

    Article  MathSciNet  MATH  Google Scholar 

  • Deschrijver, G. (2011). Triangular norms which are meet-morphisms in interval-valued fuzzy set theory. Fuzzy Sets and Systems, 181, 88–101.

    Article  MathSciNet  MATH  Google Scholar 

  • Deschrijver, G., Cornelis, C., & Kerre, E. E. (2004). On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Transactions on Fuzzy Systems, 12(1), 45–61.

    Article  MathSciNet  Google Scholar 

  • Fodor, J. C. (1993). A new look at fuzzy connectives. Fuzzy Sets and Systems, 57, 141–148.

    Article  MathSciNet  MATH  Google Scholar 

  • Gehrke, M., Walker, C., & Walker, E. (1996). Some comments on interval valued fuzzy sets. International Journal of Intelligent Systems, 11, 751–759.

    Article  MATH  Google Scholar 

  • Gorzalczany, M. B. (1987). A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets and Systems, 21, 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  • Herrera, F., Alonso, S., Chiclana, F., & Herrera-Viedma, E. (2009). Computing with words in decision making: foundations, trends and prospects. Fuzzy Optimization and Decision Making, 8(4), 337–364.

    Article  MathSciNet  Google Scholar 

  • Higashi, M., & Klir, G. J. (1982). On measure of fuzziness and fuzzy complements. International Journal of General Systems, 8(3), 169–180.

    Article  MathSciNet  MATH  Google Scholar 

  • Klement, E., & Navara, M. (1999). A survey on different traingular norm-based fuzzy logics. Fuzzy Sets and Systems, 101, 241–251.

    Article  MathSciNet  MATH  Google Scholar 

  • Li, D. C., & Li, Y. M. (2012). Algebraic structures of interval-valued fuzzy (S, N)-implications. International Journal of Approximate Reasoning, 53, 892–900.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, H. W., & Wang, G. J. (2006). A note on implicators based on binary aggregation operators in interval-valued fuzzy set theory. Fuzzy Sets and Systems, 157, 3231–3236.

    Article  MathSciNet  MATH  Google Scholar 

  • Lowen, R. (1978). On fuzzy complements. Information Sciences, 14(2), 107–113.

    Article  MathSciNet  MATH  Google Scholar 

  • Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to interval analysis. Philadelpha: SIAM.

    Book  MATH  Google Scholar 

  • Ovchinnikov, S. V. (1983). General negations in fuzzy set theory. Journal of Mathematical Analysis and Applications, 92(1), 234–239.

    Article  MathSciNet  MATH  Google Scholar 

  • Santiago, R. H. N., Bedregal, B. C., & Acióly, B. M. (2006). Formal aspects of correctness and optimality in interval computations. Formal Aspects of Computing, 18(2), 231–243.

    Article  MATH  Google Scholar 

  • Trillas, E. (1979). Sobre funciones de negación en la teoria de conjuntos difusos. Stochastica, 3, 47–59.

    MathSciNet  MATH  Google Scholar 

  • Wu, J. C., & Luo, M. K. (2011). Fixed points of involutive interval-valued negations. Fuzzy Sets and Systems, 182(1), 110–118.

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh, L. A. (1971). Quantitative fuzzy semantics. Information Science, 3, 159–176.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referees and the Editor-in-Chief for their valuable comments. This work was supported by the Natural Science Foundation of China (Grant No: 11201279), Zhejiang Provincial Natural Science Foundation (Grant No: LY12A01009), Zhejiang Province Commonweal Technique Research Project (Grant No: 2013C31001) and the Open Foundation from Marine Sciences in the Most Important Subject of Zhejiang (Grant No: 20140102).

Author information

Authors and Affiliations

  1. School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan, 316022, China

    Li Dechao & Jiang Youfu

  2. Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang Province, Zhoushan, 316022, China

    Li Dechao & Jiang Youfu

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

    Xie Yongjian

Authors
  1. Li Dechao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xie Yongjian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiang Youfu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Dechao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dechao, L., Yongjian, X. & Youfu, J. Natural negation of interval-valued t-(co) norms and implications. Fuzzy Optim Decis Making 15, 1–20 (2016). https://doi.org/10.1007/s10700-015-9214-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10700-015-9214-8

Keywords

Search

Navigation