The ⊛-composition of fuzzy implications: Closures with respect to properties, powers and families

Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications from a given pair of fuzzy implications. Viewing this as a binary operation ~ on the set I of all fuzzy implications they obtained, for the first time, a monoid structure (I,~) on the set I. Some algebraic aspects of (I,~) had already been explored and hitherto unknown representation results for the Yager’s families of fuzzy implications were obtained in [Representations through a Monoid on the set of Fuzzy Implications, Fuzzy Sets and Systems, 247, 51-67]. However, the properties of fuzzy implications generated or obtained using the ~-composition have not been explored. In this work, the preservation of the basic properties like neutrality, ordering and exchange principles, the functional equations that the obtained implications satisfy, the powers w.r.t. ~ and their convergence, and the closures of some families of fuzzy implications w.r.t. the operation ~, specifically the families of (S,N)-, R-, f and g-implications, are studied. This study shows that the ~-composition carries over many of the desirable properties of the original fuzzy implications to the generated fuzzy implications and further, due to the associativity of the ~-composition one can obtain, often, infinitely many new fuzzy implications from a single fuzzy implication through self-composition w.r.t. the ~-composition.