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Keywords:
fuzzy implications; the law of importation; the law of contra-positive symmetry; $(S;N)$-implications; $R$-implications
fuzzy implications; the law of importation; the law of contra-positive symmetry; $(S;N)$-implications; $R$-implications
Summary:
It is well known that monotonicity has been an important defining criterion for fuzzy logic connectives, such as fuzzy negations, t-norms, t-conorms and fuzzy implications. Also, a stronger version of monotonicity, namely strict monotonicity, establishes some significant representation theorems of continuous fuzzy negations, continuous t-norms and continuous t-conorms. In this work, we propose the strict monotonicity for fuzzy implications and investigate some necessary conditions on fuzzy implications to fulfill the same. Also, the relationship between the basic properties, functional equations of fuzzy implications and the strict monotonicity will be investigated. Further, we examine the strict monotonicity for fuzzy implications that do come from different families of fuzzy implications and show that the strict monotonicity is a necessary condition for fuzzy polynomial implications, fuzzy rational implications and some subclasses of $(S,N)$ and $f$-generated fuzzy implications.
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