@Article{PetrikSarkoci:ConvNilp,
  IS = { zkontrolovano 29 Sep 2009 },
  UPDATE  = { 2009-09-29 },
  author =     { Petr{\'\i}k, Milan and Sarkoci, Peter },
  title =      { Convex combinations of nilpotent triangular norms },
  year =       { 2009 },
  month =      { February },
  pages =      { 271--275 },
  journal =    { Journal of Mathematical Analysis and Applications },
  publisher =  { Elsevier },
  address =    { Amsterdam, Netherlands },
  issn =       { 0022-247X },
  volume =     { 350 },
  number =     { 1 },
  authorship = { 50-50 },
  annote = { In this paper we deal with the open problem of convex
    combinations of continuous triangular norms stated by Alsina,
    Frank, and Schweizer [C. Alsina, M.J. Frank, B. Schweizer,
    Problems on associative functions, Aequationes Math. 66 (2003)
    128--140, Problems 5 and 6]. They pose a question whether a
    non-trivial convex combination of triangular norms can ever be a
    triangular norm. The main result of this paper gives a negative
    answer to the question for any pair of continuous Archimedean
    triangular norms with different supports. With the help of this
    result we show that a non-trivial convex combination of nilpotent
    t-norms is never a t-norm. The main result also gives an
    alternative proof to the result presented by Ouyang and Fang [Y.
    Ouyang, J. Fang, Some observations about the convex combination of
    continuous triangular norms, Nonlinear Anal., 68 (11) (2008)
    3382--3387, Theorem 3.1]. In proof of the main theorem we utilize
    the Reidmeister condition known from the web geometry. },
  keywords =   { Nilpotent triangular norm, Reidmeister condition, 
                 Convex combination },
  note =       { DOI: 10.1016/j.jmaa.2008.09.060 },
  project =    { GACR 201/07/1136 },
psurl       = { [PDF, 204 KB] },
www         = { http://dx.doi.org/10.1016/j.jmaa.2008.09.060 },
}