Type-2 fuzzy implications and fuzzy-valued approximation reasoning

In this paper, the quasi-distributivity laws of type-2 fuzzy implications with respect to extended supremum and extended infimum are investigated on different subalgebras of fuzzy truth values. Moreover, the properties of fuzzy-valued fuzzy implications are further discussed. Especially, new fuzzy-valued operations are obtained from a fuzzy-valued fuzzy implication induced by a fuzzy implication, which is right-continuous with respect to the second argument. As an application of fuzzy-valued fuzzy implications, fuzzy-valued approximate reasoning is further studied and a numerical example is given.