On dependencies and independencies of fuzzy implication axioms

A fuzzy implication, commonly defined as a two-place operation on the unit interval, is an extension of the classical binary implication. It plays important roles in both mathematical and applied sides of fuzzy set theory. Besides the basic axioms, there are many potential fuzzy implication axioms, among which eight are widely used in the literature. Different fuzzy implications satisfying different subgroups of these eight axioms can be found. However, certain interrelationships exist between these eight axioms. But the results remain incomplete. This paper aims to lay bare the interrelationships between these eight axioms. The result is instrumental to propose a classification of fuzzy implications.

[1]  Humberto Bustince,et al.  Automorphisms, negations and implication operators , 2003, Fuzzy Sets Syst..

[2]  Siegfried Gottwald,et al.  Fuzzy Sets and Fuzzy Logic , 1993 .

[3]  E. Trillas,et al.  On contra‐symmetry and MPT conditionality in fuzzy logic , 2005, Int. J. Intell. Syst..

[4]  J. Drewniak,et al.  Monotonic Fuzzy Implications , 2000 .

[5]  Michal Baczynski,et al.  Fuzzy Implications , 2008, Studies in Fuzziness and Soft Computing.

[6]  E. Kerre.  A call for crispness in fuzzy set theory , 1989 .

[7]  Etienne Kerre,et al.  Fuzzy techniques in image processing , 2000 .

[8]  Ronald R. Yager,et al.  On some new classes of implication operators and their role in approximate reasoning , 2004, Inf. Sci..

[9]  E. Trillas.  Sobre funciones de negación en la teoría de conjuntos difusos. , 1979 .

[10]  Michal Baczynski,et al.  Residual implications revisited. Notes on the Smets-Magrez Theorem , 2004, Fuzzy Sets Syst..

[11]  Balasubramaniam Jayaram,et al.  Rule reduction for efficient inferencing in similarity based reasoning , 2008, Int. J. Approx. Reason..

[12]  Etienne Kerre,et al.  Fuzzy implication operators and generalized fuzzy method of cases , 1993 .

[13]  Joan Torrens,et al.  A Survey on Fuzzy Implication Functions , 2007, IEEE Transactions on Fuzzy Systems.

[14]  Humberto Bustince,et al.  Construction of fuzzy indices from fuzzy DI-subsethood measures: Application to the global comparison of images , 2007, Inf. Sci..

[15]  Guoqing Chen,et al.  Discovering a cover set of ARsi with hierarchy from quantitative databases , 2005, Inf. Sci..

[16]  J. Fodor.  Contrapositive symmetry of fuzzy implications , 1995 .

[17]  George J. Klir,et al.  Fuzzy sets and fuzzy logic - theory and applications , 1995 .

[18]  Paulo J. G. Lisboa,et al.  Fuzzy systems in medicine , 2000 .

[19]  George J. Klir,et al.  Fuzzy sets and fuzzy logic , 1995 .

[20]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[21]  Etienne E. Kerre,et al.  Fuzzy If-Then Rules in Computational Intelligence: Theory and Applications , 2012 .

[22]  Etienne Kerre,et al.  Fuzzy Adjunctions in Mathematical Morphology , 2003 .

[23]  Daowu Pei,et al.  R0 implication: characteristics and applications , 2002, Fuzzy Sets Syst..

[24]  Wen-Xiu Zhang,et al.  Hybrid monotonic inclusion measure and its use in measuring similarity and distance between fuzzy sets , 2009, Fuzzy Sets Syst..

[25]  Józef Drewniak,et al.  Invariant Fuzzy Implications , 2006, Soft Comput..