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Fibred triangular norms. (English) Zbl 0946.26017

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:

26E50 Fuzzy real analysis
03E72 Theory of fuzzy sets, etc.
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References:

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