@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 }, }