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Zbl 0946.26017
Jenei, Sándor
Fibred triangular norms. (English)
[J] Fuzzy Sets Syst. 103, No.1, 67-82 (1999). ISSN 0165-0114

Summary: We deal with the formula $g[T(f(x), f(y))]$ where $f$, $g$ are functions mapping the unit interval onto itself and $T$ is a triangular norm. We are interested in the conditions under which this formula yields a triangular norm. The case, when $f$ is continuous is investgated in detail. As a result, a new method which generates new triangular norms from an arbitrary triangular norm is developed. These new triangular norms are the so-called `fibred triangular norms' which are closely related to the homomorphism of semigroups. It is well known that continuous Archimedean triangular norms can be generated with additive generator functions. We investigate a possible generalization of the additive generator function and characterize the class of t-norms which can be generated with them. This class turns out to be the class of fibred continuous Archimedean t-norms. Finally, some examples are given for the case when $f$ is not continuous.

MSC 2000:: *26E50 Fuzzy real analysis
03E72 Fuzzy sets (logic)

Keywords: Łukasiewicz t-norm; triangular norms; homomorphism; semigroups; additive generator function; t-norms; fibred continuous Archimedean t-norms

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