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In this short note, we provide a continuous triangular norm that is not a copula and that generates a metric. Thus, the 9th open problem collected in Alsina et al. [Associative Functions: Triangular Norms and Copulas, World Scientific Publishing Company, Singapore, 2006] has a negative answer.
First, a characterization is given for the continuity of a class of left-continuous triangular norms: it is established that a left-continuous triangular norm with strong induced negation is continuous if and only if it is strictly increasing on the domain where its value is positive. Second, we show that axioms of a left-continuous triangular norm with strong induced negation are not independent,...
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