Generators of triangular norms

H-transformation of t-norms

“…There are only few known classes of t-norms with anomalous pairs, see [16]. New light into the construction of t-norms with anomalous pairs brings the example of Hájek in [7], which we modify into a new construction method for t-norms with anomalous pairs.…”

Cancellativity properties for t-norms and t-subnorms

“…Conversely, a cancellative t-norm which is continuous at the point ð1; 1Þ will be continuous if and only if it has no anomalous pair (i.e., there do not exist x; y 2 0; 1½ for which x > y > x ð2Þ T > y ð2Þ T > Á Á Á > x ðnÞ T > y ðnÞ T > Á Á Á, with n 2 N n f0g) [17]. It has been shown in [1] that there also exist cancellative, non-continuous t-norms that are left-continuous.…”

“…For a more profound study of both types of cancellativity in terms of additive generators of t-norms we refer to [17].…”

Semi-divisible triangular norms

“…(4). To stress the difference between t-norms and t-subnorms, recall a related result that a continuous Archimedean t-subnorm need not to be generated, see (Mesiarová 2005 …”