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Keywords:
strict monotonicity; aggregation functions; fuzzy implications; fuzzy negations; lattice operations; convex combinations; conjugacy classes
strict monotonicity; aggregation functions; fuzzy implications; fuzzy negations; lattice operations; convex combinations; conjugacy classes
Summary:
Recently, in [20], the authors proposed the strict monotonicity for fuzzy implications and investigated various aspects of fuzzy implications such as properties, functional equations and families of fuzzy implications with respect to strict monotonicity. In this work, the preservation of strict monotonicity is examined for various generating methods of fuzzy implications that are available in the literature. More precisely, the generating methods such as the $N$-reciprocal method, the conjugacy classes, the aggregation of fuzzy implications, the sup -$T$ composition, the lattice operations, the convex combinations, the $ \triangledown$-composition, the $\circledast$-composition, the $e$-threshold generating method and the vertical $e$-threshold generating methods are considered and the conditions under which each of these methods preserve the strict monotonicity are investigated. Further, we investigate the strict monotonicity of some new classes of fuzzy implications proposed in the context of generalized hypothetical syllogism and prove that the aggregation of fuzzy implications coming from these classes satisfy the strict monotonicity always. As an interesting outcome of this study, we show that strict monotonicity is a criterion to characterize whether a given fuzzy implication is self-conjugate or not.
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