n-Dimensional fuzzy implications: analytical, algebraic and applicational approaches - Zanotelli, Rosana Medina (2020)
The n-dimensional fuzzy logic (n-DFL) is an extension of the fuzzy logic (FL) as old as hesitant fuzzy sets and less exploited, motivating new investigations, and promoting results to consolidate this research area. The study of n-DFL contributes to overcome the insufficiency of traditional fuzzy logic in modeling imperfect and imprecise information coming from different opinions of experts. Moreover, the possibility to model repeated and ordered degrees of membership in the n-dimensional fuzzy sets is considered a consolidated strategy in applied technologies including areas as pattern recognition, image processing, data mining and mathematical morphology. This large field of applications motivate the studies developed in this work. Based on representability of n-dimensional fuzzy connectives, we are able to extend relevant theoretical results from fuzzy connectives to n-dimensional fuzzy approach. In particular, this proposal introduces the study of n-dimensional fuzzy implications (n-DI) following distinct approaches: (i) analytical studies, defining n-DI, presenting the most desirable properties as neutrality, ordering, (contra-)symmetry, exchange and identity principles, and also discussing their interrelationships and exemplifications; (ii) algebraic aspects mainly related to left- and right-continuity of representable n-dimensional fuzzy t-norms and the generation of n-DI from existing fuzzy implications; (iii) n-dimensional approach of fuzzy implication classes explicitly represented by fuzzy connectives, as (S;N)-implications and QL-implications; (iv) prospective studies of n-dimensional R-implications (n-DRI), analyzing extended conditions to verify the residuation principle and their characterization based on n-dimensional t-norms and n-DI; (v) constructive method obtaining n-DRI based on n-dimensional aggregation operators, presenting an exemplification in the solution of a problem involving CIM-MCDM, based on Łukasiewicz n-DRI; and also including (vi) an introductory study considering an n-DI in modeling approximate reasoning of inference schemes, dealing with the effecting role of such connectives in based-rule fuzzy systems. In addition, taking into account the action of automorphism and fuzzy negations on Ln(U), conjugate and dual operators of n-DI can be respectively obtained, preserving algebraic and analytical properties.