K-operators: An approach to the generation of interval-valued fuzzy implications from fuzzy implications and vice versa

This paper introduces the interval-valued fuzzy implications generated from fuzzy implications and from K-operators showing that this construction generalizes the canonical representation of fuzzy implications. Such generalization allows to recover the usual fuzzy implication properties when dealing with interval-valued fuzzy implications. In addition, the condition under which the converse construction, i.e. the generation of fuzzy implications by action of K-operators and interval-valued fuzzy implications, is discussed. Moreover, we analysed their conjugate construction obtained by action of interval automorphisms. Through commutative diagrams, the associativity related to compositions of conjugate, dual and K-operators performed over fuzzy implications is shown, also preserving their main properties.