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Contrapositive symmetry of fuzzy implications. (English) Zbl 0845.03007
Summary: Contrapositive symmetry of $$R$$- and $$QL$$-implications defined from $$t$$-norms, $$t$$-conorms and strong negations is studied. For $$R$$-implications, characterizations of contrapositive symmetry are proved when the underlying $$t$$-norm satisfies a residuation condition. Contrapositive symmetrization of $$R$$-implications not having this property makes it possible to define a conjunction so that the residuation principle is preserved. Cases when this associated conjunction is a $$t$$-norm are characterized. As a consequence, a new family of $$t$$-norms (called nilpotent minimum) owing several attractive properties is discovered. Concerning $$QL$$-implications, contrapositive symmetry is characterized by solving a functional equation. When the underlying $$t$$-conorm is continuous and the $$t$$-norm is Archimedean, the $$t$$-conorm must be isomorphic to the Lukasiewicz one, while the $$t$$-norm must be isomorphic to a member from the well-known Frank family of $$t$$-norms. Finally, contrapositive symmetry for some new families of fuzzy implications is investigated.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness
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