NON–CONTINUOUS t –NORMS WITH CONTINUOUS DIAGONAL

Dana Smutná

   During the workshops at the 2nd International Conference on Fuzzy Sets Theory and Its Applications (Liptovský Mikuláš, Slovak Republic, January 31-February 4, 1994), several open problems were specified by the participants, see [4]. One of them was the question: If T is an Archimedean t -norm with a continuous diagonal, is T necessarily continuous on [0,1]×[0,1]? (This problem was stated in the book of Schweizer and Sklar [5] and Kimberling [1] has given an example of a continuous Archimedean t -norm which is not uniquely determined by its diagonal.) However, the above problem was already negatively solved by Gerianne Krause, but not yet published. In the paper we show a new class of Archimedean, non-continuous t -norms with a continuous diagonal.

Keywords: triangular norms, multiplicative generator, non-continuous Archimedean t -norms

[full-paper]