Q L-operations and Q L-implication functions constructed from tuples (O, G, N) and the generation of fuzzy subsethood and entropy measures

Considering the important role played by overlap and grouping functions in several applications in which associativity is not demanded, in this paper we introduce the notion of Q L-operations constructed from tuples (O, G, N), where overlap functions O, grouping functions G and fuzzy negations N are used for the generalization of the implication p -> q (-)p boolean OR (p boolean AND q), which is defined in quantum logic (Q L). We also study under which conditions Q L-operations constructed from tuples (O, G, N) are fuzzy implication functions, presenting a general form for obtaining Q L-implication functions, and particular forms of such fuzzy implication functions according to specific properties of O and G. We analyze the main properties satisfied by Q L-operations and Q L-implication functions, establishing under which conditions of O, G and N, the derived Q L-operations (implication functions) satisfy the different known properties for fuzzy implication functions. We show that Q L-implication functions constructed from tuples (O, G, N) are richer than Q L-implication functions constructed from t-norms and positive t-conorms. We provide a comparative study of Q L-implication functions and other classes of fuzzy implication functions constructed from fuzzy negations, overlap and grouping functions, analyzing the intersections among such classes. Finally, we present the application of both Q L-operations and Q L-implication functions constructed from tuples (O, G, N) to the generation of fuzzy subsethood and derived entropy measures, which are useful for several practical applications. (C) 2017 Elsevier Inc. All rights reserved.