The circle star-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 circle star on the set I of fuzzy implications they obtained, for the first time, a monoid structure (I, circle star) on the set I. Some algebraic aspects of (I, circle star) 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) 51-67). However, the properties of fuzzy implications generated or obtained using the circle star-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. circle star and their convergence, and the closures of some families of fuzzy implications w.r.t. the operation circle star, specifically the families of (S, N)-, R-, f-and g-implications, are studied. This study shows that the circle star-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 circle star-composition one can obtain, often, infinitely many new fuzzy implications from a single fuzzy implication through self-composition w.r.t. the circle star-composition