-Dimensional (,)-implications.

The -dimensional fuzzy logic (-DFL) has been contributed to overcome the insufficiency of traditional fuzzy logic in modeling imperfect and imprecise information, coming from different opinions of many experts by considering the possibility to model not only ordered but also repeated membership degrees. Thus, -DFL provides a consolidated logical strategy for applied technologies since the ordered evaluations provided by decision makers impact not only by selecting the best solutions for a decision making problem, but also by enabling their comparisons. In such context, this paper studies the -dimensional fuzzy implications (-DI) following distinct approaches: (i) analytical studies, presenting the most desirable properties as neutrality, ordering, (contra-)symmetry, exchange and identity principles, discussing their interrelations and exemplifications; (ii) algebraic aspects mainly related to left- and right-continuity of representable -dimensional fuzzy t-conorms; and (iii) generating -DI from existing fuzzy implications. As the most relevant contribution, the prospective studies in the class of -dimensional interval (S,N)-implications include results obtained from t-representable -dimensional conorms and involutive -dimensional fuzzy negations. And, these theoretical results are applied to model approximate reasoning of inference schemes, dealing with based-rule in -dimensional interval fuzzy systems. A synthetic case-study illustrates the solution for a decision-making problem in medical diagnoses.