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On some geometric transformation of \(t\)-norms. (English) Zbl 0932.03066
Summary: Given a triangular norm \(T\), its t-reverse \(T^*\), introduced by C. Kimberling [Publ. Math., Debrecen 20, 21-39 (1973; Zbl 0276.26011)] under the name of invert, is studied. The question under which conditions we have \(T^{**}= T\) is completely solved. The t-reverses of ordinal sums of t-norms are investigated and a complete description of continuous, self-reverse t-norms is given, leading to a new characterization of the continuous t-norms \(T\) such that the function \(G(x,y)= x+ y- T(x,y)\) is a t-conorm, a problem originally studied by M. J. Frank [Aequationes Math. 19, 194-226 (1979; Zbl 0444.39003)]. Finally, some open problems are formulated.

MSC:
03E72 Theory of fuzzy sets, etc.
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