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 I of fuzzy implications they obtained, for the first time, a monoid structure (I,?) ( I , ? ) on the set I I . Some algebraic aspects of (I,?) ( I , ? ) had already been explored and hitherto unknown representation results for the Yager's families of fuzzy implications were obtained in [53] (N.R. Vemuri and B. Jayaram, Representations through a monoid on the set of fuzzy implications, fuzzy sets and systems, 247 (2014) 5167). 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 fuzzy 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) ( 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.
fuzzy implication closure basic properties functional equations self-composition (sn)-implications r-implications f-implications g-implications