A new class of fuzzy implications. Axioms of fuzzy implication revisited

Many different fuzzy implication operators have been proposed; most of them fit into one of the two classes: implication operations that are based on an explicit representation of implication A --> B in terms of &, V, and inverted left perpendicular (e.g., S-implications that are based on the formula B V inverted left perpendicular A), and R-implications that are based on an implicit representation of implication A --> B as the weakest C for which C & B implies A. However, some fuzzy implication operations (such as b(a)) cannot be naturally represented in this form. To describe such operations, we propose a new (third) class of implication operations called A-implications whose relation to &, V, and inverted left perpendicular is described by (implicit) axioms. (C) 1998 Published by Elsevier Science B.V. All rights reserved.