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| - 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.
- 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. (en)
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Title
| - Convex combinations of nilpotent triangular norms
- Convex combinations of nilpotent triangular norms (en)
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skos:prefLabel
| - Convex combinations of nilpotent triangular norms
- Convex combinations of nilpotent triangular norms (en)
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skos:notation
| - RIV/68407700:21230/09:00157777!RIV10-GA0-21230___
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http://linked.open...avai/riv/aktivita
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http://linked.open...avai/riv/aktivity
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http://linked.open...ai/riv/idVysledku
| - RIV/68407700:21230/09:00157777
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http://linked.open...riv/jazykVysledku
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http://linked.open.../riv/klicovaSlova
| - Nilpotent triangular norm; Reidmeister condition; Convex combination (en)
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http://linked.open.../riv/klicoveSlovo
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http://linked.open...odStatuVydavatele
| - US - Spojené státy americké
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| - Journal of Mathematical Analysis and Its Applications
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