Skip to main content
SpringerLink
Log in

Convex combinations of strict t-norms

  • Focus
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Alsina, Frank, and Schweizer posed a question whether a non-trivial convex combination of triangular norms can ever be a triangular norm. In this paper we deal with the class of strict t-norms with well-defined first partial derivatives along their zero borders. We show that the answer to the question is negative for certain couples of such t-norms and we state out possible further research.

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

  • Abel N (1826) Untersuchungen der Funktionen zweier unabhängigen veränderlichen Grössen x und y wie f(x,y), welche die Eigenschaft von x, y und z ist. Journal für die reine und angewandte Mathematik 1:11–15

    Article  MATH  Google Scholar 

  • Aczél J (1949) Sur les opérations definies pour des nombres réels. Bulletin de la Société Mathématique de France 76:59–64

    Google Scholar 

  • Alsina C (1992) On a method of Pi-Calleja for describing additive generators of associative functions. Aequationes Mathematicae 43:14–20

    Article  MATH  MathSciNet  Google Scholar 

  • Alsina C, Frank MJ, Schweizer B (2003) Problems on associative functions. Aequationes Mathematicae 66(1–2):128–140

    MATH  MathSciNet  Google Scholar 

  • Alsina C, Frank MJ, Schweizer B (2006) Associative functions: triangular norms and copulas. World Scientific, Singapore

    Book  MATH  Google Scholar 

  • Chalice DR (1991) A Characterization of the Cantor function. Amer Math Monthly 98:255-258

    Article  MATH  MathSciNet  Google Scholar 

  • Durante F, Sarkoci P (2008) A note on the convex combinations of triangular norms. Fuzzy Sets Syst 159(1):77–80. doi:10.1016/j.fss.2007.07.005

    Article  MATH  MathSciNet  Google Scholar 

  • Faucett WM (1955) Compact semigroups irreducibly connected between two idempotents. Proc Am Math Soc 6:741–747

    Article  MATH  MathSciNet  Google Scholar 

  • Jenei S (2006) On the convex combination of left-continuous t-norms. Aequationes Mathematicae 72(1–2):47–59

    Article  MATH  MathSciNet  Google Scholar 

  • Klement EP, Mesiar R, Pap E (2000) Triangular norms, vol. 8 of trends in logic. Kluwer Academic Publishers, Dordrecht, Netherlands

  • Ling CM (1965) Representation of associative functions. Publicationes Mathematicae Debrecen 12:189–212

    MathSciNet  Google Scholar 

  • Lowen R (1996) Fuzzy set theory. Basic concepts, techniques, and bibliography. Kluwer Academic Publishers, Dordrecht, Netherlands

    MATH  Google Scholar 

  • Mesiar R, Mesiarová-Zemánková A (2008) Convex combinations of continuous t-norms with the same diagonal function. Nonlinear Anal Theory Methods Appl 69(9):2851–2856

    Article  MATH  Google Scholar 

  • Mostert PS, Shields AL (1957) On the structure of semigroups on a compact manifold with boundary. Ann Math 65:117–143

    Article  MathSciNet  Google Scholar 

  • Navara M, Petrík M (2008) Two methods of reconstruction of generators of continuous t-norms. In: Magdalena L, Ojeda-Aciego M, Verdegay JL (eds) IPMU 2008: information processing and management of uncertainty in knowledge-based systems, Málaga, Spain, pp 1016–1021. Department of Applied Mathematics, University of Málaga, CD

  • Navara M, Petrík M, Sarkoci P (2009) Explicit formulas for generators of triangular norms. Submitted

  • Ouyang Y, Fang J (2008) Some observations about the convex combinations of continuous triangular norms. Nonlinear Anal Theory Methods Appl 68(11):3382–3387

    Article  MATH  MathSciNet  Google Scholar 

  • Ouyang Y, Fang J, Li G (2007) On the convex combination of T D and continuous triangular norms. Inf Sci 177(14):2945–2953

    Article  MATH  MathSciNet  Google Scholar 

  • Petrík M (2009) Properties of fuzzy logical operations. Ph.D. thesis, CTU - FEE, Prague, June 2009

  • Petrík M, Sarkoci P (2009) Convex combinations of nilpotent triangular norms. J Math Anal Appl 350:271–275. doi:10.1016/j.jmaa.2008.09.060

    Article  MATH  MathSciNet  Google Scholar 

  • Pi-Calleja P (1954) Las ecuacionas funcionales de la teoría de magnitudes. Segundo Symposium de Matemática, Villavicencio, Mendoza, Coni, Buenos Aires, pp 199–280

  • Schweizer B, Sklar A (2006) Probabilistic metric spaces, 2nd edn. Dover Publications, Mineola, NY. North-Holland, Amsterdam 1983

  • Tomás MS (1987) Sobre algunas medias de funciones asociativas. Stochastica XI(1):25–34

    Google Scholar 

  • Vetterlein T (2008) Regular left-continuous t-norms. Semigroup Forum 77(3):339–379

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author would like to thank the anonymous referee for valuable comments. The author was supported by Czech Science Foundation under Project 201/07/1136.

Author information

Authors and Affiliations

  1. Department of Cybernetics, Faculty of Electrical Engineering, Center for Machine Perception, Czech Technical University, Technická 2, 166 27, Prague 6, Czech Republic

    Milan Petrík

Authors
  1. Milan Petrík
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Milan Petrík.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Petrík, M. Convex combinations of strict t-norms. Soft Comput 14, 1053–1057 (2010). https://doi.org/10.1007/s00500-009-0484-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-009-0484-3

Keywords

Search

Navigation