Generalized extended fuzzy implications

This paper is devoted to investigate generalized extended fuzzy implications in accordance with the generalized extension principle. The algebraic properties of generalized extended fuzzy implications are discussed on different algebras of fuzzy truth values. On this basis, fuzzy-valued fuzzy implications are constructed from arbitrary fuzzy implications and t-norms. Moreover, a fuzzy-valued fuzzy implication is identical with a generalized extended fuzzy implication on the algebra of fuzzy values, when they are induced by exactly the same continuous fuzzy implication and same upper semicontinuous t-norm.